Group von Neumann Algebras
and Related Algebras
Dissertation
zur Erlangung des Doktorgrades
der Mathematisch-Naturwissenschaftlichen Fachbereiche
der Georg-August-Universität zu Göttingen
vorgelegt von
Holger Reich
aus Uelzen
Göttingen 1998
D7
Referent: Prof. Dr. W. Lück
Korreferent: Prof. Dr. T. tom Dieck
Tag der mündlichen Prüfung:
Contents
1 Introduction
1.1 Notations and Conventions . . . . . . . . . . . . . . . . . . . . .
1
4
2 The Algebra of Operators Affiliated to a Finite von Neumann
Algebra
5
3 Dimensions
10
3.1 A-modules, U-modules and Hilbert A-modules . . . . . . . . . . 10
3.2 A Notion of Dimension for U-modules . . . . . . . . . . . . . . . 15
3.3 The Passage from A-modules to U-modules . . . . . . . . . . . . 19
4 New Interpretation of L2 -Invariants
24
5 The
5.1
5.2
5.3
26
26
29
32
Atiyah Conjecture
The Atiyah Conjecture . . . . . . . . . . . . . . . . . . . . . . . .
A Strategy for the Proof . . . . . . . . . . . . . . . . . . . . . . .
The Relation to the Isomorphism Conjecture in Algebraic K-theory
6 Atiyah’s Conjecture for the Free Group on Two Generators
37
6.1 Fredholm Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2 Some Geometric Properties of the Free Group . . . . . . . . . . . 39
6.3 Construction of a Fredholm Module . . . . . . . . . . . . . . . . 40
7 Classes of Groups and Induction Principles
43
7.1 Elementary Amenable Groups . . . . . . . . . . . . . . . . . . . . 43
7.2 Linnell’s Class C . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8 Linnell’s Theorem
8.1 Induction Step: Extensions - The (A)-Part . . .
8.2 Induction Step: Extensions - The (B)-Part . . . .
8.3 Induction Step: Directed Unions . . . . . . . . .
8.4 Starting the Induction - Free Groups . . . . . . .
8.5 Starting the Induction - Finite Extensions of Free
9 Homological Properties and Applications
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Groups
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10 K-theory
81
10.1 K0 and K1 of A and U . . . . . . . . . . . . . . . . . . . . . . . . 81
10.2 Localization Sequences . . . . . . . . . . . . . . . . . . . . . . . . 85
11 Appendix I: Affiliated Operators
87
12 Appendix II: von Neumann Regular Rings
95
i
13 Appendix III: Localization of Non-commutative Rings
13.1 Ore Localization . . . . . . . . . . . . . . . . . . . . . . .
13.2 Universal Localization . . . . . . . . . . . . . . . . . . . .
13.3 Division Closure and Rational Closure . . . . . . . . . . .
13.4 Universal R-Fields . . . . . . . . . . . . . . . . . . . . . .
13.5 Recognizing Universal Fields of Fractions . . . . . . . . .
13.6 Localization with Respect to Projective Rank Functions .
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98
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14 Appendix IV: Further Concepts from the Theory of Rings
111
14.1 Crossed Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
14.2 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
ii
1
Introduction
In their first Rings of Operators paper from 1936 Murray and von Neumann
not only introduced those algebras which are nowadays known as von Neumann
algebras but they also considered for the first time a notion of dimension which
is not necessarily integer valued [63]. Since then this idea has found a lot of
applications. In particular in 1976, studying indices of elliptic operators on
coverings, Atiyah was led to define the so called L2 -Betti numbers which are a
priori real valued [1]. It turned out later that these numbers are in fact topological and even homotopy invariants [24]. Meanwhile they admit of a definition
completely analogous to the usual Betti numbers [52]. The only difference is
that whereas usual Betti numbers are ranks of Z-modules the L2 -Betti numbers
are dimensions of modules over the group von Neumann algebra N Γ. Here Γ
can for example be the fundamental group.
In order to compute the usual Betti numbers it is often easier to pass to rational
homology and to compute vector space dimensions. In this thesis we investigate
a similar passage for von Neumann algebras. The algebra of operators UΓ
affiliated to the von Neumann algebra N Γ, which was already introduced by
Murray and von Neumann [63], plays the role of a quotient field for N Γ. We
will show that there is a notion of dimension for arbitrary UΓ-modules which is
compatible with the N Γ-dimension developed by Lück in [52].
Moreover, the algebras UΓ are large enough to host similar quotient fields DΓ
for the group algebras CΓ. Linnell made use of this idea to prove refined versions
of the zero divisor conjecture for a large class of groups [48]. It also leads to
a natural explanation of the fact that in all known cases the L2 -Betti numbers
are rational numbers. Investigating these intermediate rings we show a kind
of universal coefficient theorem for L2 -homology and obtain information about
Euler characteristics of groups. We also try to clarify the relationship to the
Isomorphism Conjecture in algebraic K-theory [27].
We will now explain the results in greater detail. Let Γ be a group and let CΓ
be the complex group ring. We investigate the following commutative square of
rings, where all maps are unit-preserving embeddings of subrings.
CΓ
- NΓ
?
DΓ
?
- UΓ
One should think of the upper horizontal map as a completion and of the vertical
maps as localizations. We start at the upper right corner: Let l2 Γ be a complex
Hilbert space with orthonormal basis the set Γ. This Hilbert space carries a
natural Γ-action and this yields an embedding of the complex group ring CΓ
into the algebra B(l2 Γ) of bounded linear operators.
1
Definition 1.1. The group von Neumann algebra N Γ is the closure of CΓ in
B(l2 Γ) with respect to the weak operator topology.
We consider N Γ as a ring extension of CΓ. In particular we can study CΓmodules by applying the functor −⊗CΓ N Γ in order to treat them in the category
of N Γ-modules. This may seem unwise at first glance, but we even go a step
further and consider a larger ring containing also unbounded operators.
Definition 1.2. The algebra UΓ of operators affiliated to the group von Neumann algebra N Γ consists of all closed, densely defined (unbounded) operators
which commute with the right action of Γ on l2 Γ.
For details see Section 2 and Appendix I. The algebra UΓ is an Ore localization
of N Γ, see Proposition 2.8. From the ring theoretical point of view it is a
beautiful ring, namely it is a von Neumann regular ring (see 2.4), i.e. its Tordimension vanishes. In particular the category of finitely generated projective
UΓ-modules is abelian. In Theorem 3.12, we show the following.
Theorem 1.3. There is a well-behaved notion of dimension for arbitrary UΓmodules which is compatible with the N Γ-dimension studied in [52].
The advantage of working with modules in the algebraic sense in contrast to
Hilbert spaces is that all tools and constructions from (homological) algebra,
like for example spectral sequences or arbitrary limits and colimits, are now
available. The L2 -Betti numbers can be read from the homology with twisted
coefficients in UΓ, see Proposition 4.2. More generally we will show that the
natural map
K0 (N Γ) → K0 (UΓ)
is an isomorphism, see Theorem 3.8. Finer invariants like the Novikov-Shubin
invariants (see [67], [25], [54]) do not survive the passage from N Γ to UΓ.
Twisted homology with coefficients in UΓ is therefore the algebraic analogue
of the reduced L2 -homology in the Hilbert space set-up, where the homology
groups are obtained by taking the kernel modulo the closure of the image of a
suitable chain complex of Hilbert spaces.
One of the main open conjectures about L2 -homology goes back to a question
of Atiyah in [1].
Conjecture 1.4 (Atiyah Conjecture). The L2 -Betti numbers of the universal covering of a compact manifold are all rational numbers.
More precisely, these numbers should be related to the orders of finite subgroups
of the fundamental group, see 5.1. We will give an algebraic reformulation of
this conjecture in Section 5 and work out how it is related to the isomorphism
conjecture in algebraic K-theory [27].
In [48] Linnell proves this conjecture for a certain class C of groups which contains free groups and is closed under extensions by elementary amenable groups.
He examines an intermediate ring DΓ of the ring extension CΓ ⊂ UΓ.
2
Definition 1.5. Let DΓ be the division closure of CΓ in UΓ, i.e. the smallest
division closed intermediate ring, compare Definition 13.14.
Linnell shows that for groups in the class C which have a bound on the orders of
finite subgroups these rings are semisimple. We will examine his proof in detail
in Section 6 and Section 8. We will emphasize that for groups in the class C
the ring DΓ is a localization of CΓ, see Theorem 8.3 and Theorem 8.4. But
for non-amenable groups one has to replace the Ore localization by a universal
localization in the sense of Cohn, compare Appendix III. Contrary to Ore localization universal localization need not be an exact functor. In Theorem 9.1, we
will show the following result.
Theorem 1.6. For groups in the class C with a bound on the orders of finite
subgroups we have
TorCΓ
p (−; DΓ) = 0
for
p ≥ 2.
This leads to a universal coefficient theorem for L2 -homology (Corollary 9.2)
and to the following result, compare Corollary 9.3.
Corollary 1.7. The group Euler characteristic for groups in the class C with a
bound on the orders of finite subgroups is nonnegative.
In Section 7 we will investigate the class C and show that it is closed under
certain processes, e.g. under taking free products. For instance P SL2 (Z) belongs
to the class C. In Section 10 we will collect results about the algebraic K-theory
of our rings CΓ, DΓ, N Γ and UΓ. In particular we compute K0 of the category
of finitely presented N Γ-modules, see Proposition 10.10. Several Appendices
which collect material mostly from ring theory will hopefully make the results
more accessible to readers with a different background.
To give the reader an idea we will now examine the easiest non-trivial examples.
Example 1.8. Let Γ = Z be an infinite cyclic group. We can identify the
Hilbert space l2 Γ via Fourier transformation with the space L2 (S1 , µ) of square
integrable functions on the unit circle S1 with respect to the usual measure. If we
think of S1 as embedded in the complex plane we
identify the group algebra
can
CZ with the algebra of Laurent polynomials C z ±1 considered as functions on
S1 . A polynomial operates on L2 (S1 , µ) by multiplication. The von Neumann
algebra can be identified with the algebra L∞ (S1 , µ) of (classes of) essentially
bounded functions. The algebra of affiliated operators is in this case the algebra
L(S1 ) of (classes of) all measurable functions. The division closure of C z ±1
in L(S1 ) is the field C(z) of rational functions. The diagram on page 1 becomes
in this case
C z ±1 - L∞ (S1 , µ)
?
C(z)
?
- L(S1 ).
3
Note that the inverse of a Laurent polynomial which has a zero on S1 is
not
bounded, but of course it is a measurable function. We obtain C(z) from C z ±1
by inverting all non-zerodivisors, i.e. all nontrivial elements.
Example 1.9. Let Γ = Z∗Z be the free group on two generators. This example
is already much more involved. It will turn out that DΓ again is a skew field.
The proof of this fact uses Fredholm module techniques and will be presented in
Section 6, compare also Theorem 5.16. This time DΓ is a universal localization
(and even a universal field of fractions) of CΓ in the sense of Cohn [15], compare
Appendix III. The functor − ⊗CΓ DΓ is no longer exact.
1.1
Notations and Conventions
All our rings are associative and have a unit. Ring homomorphisms are always
unit-preserving. For a ring R we denote by R× the group of invertible elements
in the ring. M(R) denotes the set of matrices of arbitrary (but finite) size.
GL(R) is the set of invertible matrices. We usually work with right-modules
since these have the advantage that matrices which represent right linear maps
between finitely generated free modules are multiplied in the way the author
learned as a student. An idempotent in a ring R is an element e with e2 = e.
We talk of projections only if the ring is a ∗-ring. They are by definition elements
p in the ring with p = p2 = p∗ . A non-zerodivisor inS
a ring R is an element which
is neither a left- nor a right-zerodivisor. A union i∈I Mi is called directed if
for i and j in I there always exists a k ∈ I such that Mi ∪ Mj ⊂ Mk . More
notations can be found in the glossary on page 115.
4
2
The Algebra of Operators Affiliated to a Finite von Neumann Algebra
We will now introduce one of our main objects of study: The algebra UΓ of
operators affiliated to the group von Neumann algebra N Γ. We will show that
UΓ is a von Neumann regular ring and that it is a localization of N Γ. Even
though we are mainly interested in these algebras associated to groups it is
convenient to develop a large part of the following more generally for finite
von Neumann algebras. Let B(H) be the algebra of bounded operators on the
Hilbert space H. A von Neumann algebra is a ∗-closed subalgebra of B(H)
which is closed with respect to the weak (or equivalently the strong) operator
topology. The famous double commutant theorem of von Neumann says that
one could also define a von Neumann algebra as a ∗-closed subalgebra with the
property A00 = A. Here M 0 for a subset M ⊂ B(H) is the commutant
M 0 = {a ∈ B(H) | am = ma for all m ∈ M }.
Recall that a von Neumann algebra A is finite if and only if there is a normal
faithful trace, i.e. a linear function trA : A → C, so that trA (ab − ba) = 0
and trA (a∗ a) = 0 implies a = 0. Normality means that the trace is continuous
with respect to the ultraweak topology. Equivalently given an increasing net
pλ of projections limλ trA (pλ ) = trA (limλ pλ ). This is the non-commutative
analogue of the monotone convergence theorem in integration theory. Note that
an increasing net of projections pλ converges strongly to the projection onto
the closure of the subspace generated by the pλ (H), and since a von Neumann
algebra is strongly closed this projection lies in A. This continuity property
of the trace will be crucial later, when we develop a dimension function for
arbitrary U-modules. Note that the trace is not unique. However if A = N Γ is
a group von Neumann algebra we will always use the trace given by a 7→< ae, e >
where e ∈ Γ ⊂ l2 Γ is the unit of Γ.
Definition 2.1. Given a finite von Neumann algebra A we denote by U be the
set of all (unbounded) operators a = (a, dom(a)) which satisfy the following
three conditions:
(i) a is densely defined, i.e. dom(a) is dense in H.
(ii) a is closed, i.e. its graph is closed in H ⊕ H.
(iii) a is affiliated, i.e. for every operator b ∈ A0 we have
ba ⊂ ab.
If A = N Γ is a group von Neumann algebra we write UΓ instead of U.
Here c ⊂ d means that the domain of the operator c is contained in the domain of
d, and restricted to this smaller domain the two coincide. For more information
on bounded and unbounded operators we refer to Appendix I. Note that an
5
operator a ∈ U which is bounded lies in A by the double commutant theorem.
Therefore A is a subset of U. We define the sum and product of two operators
a, b ∈ U as the closure of the usual sum and product of unbounded operators.
It is not obvious that these closures exist and lie in U, but in fact even much
more holds.
Theorem 2.2. The set U together with these structures is a ∗-algebra which
contains the von Neumann algebra A as a ∗-subalgebra.
Proof. This is already proven in the first Rings of Operators paper by Murray
and von Neumann [63]. We reproduce a proof in Appendix I.
Example 2.3. Let X ⊂ R be a closed subset. Let A = L∞ (X, µ) be the von
Neumann algebra of essentially bounded functions on X with respect to the
Lebesgue measure µ. This algebra acts on the Hilbert space L2 (X, µ) of square
integrable functions by multiplication. The associated algebra U of affiliated
operators can be identified with the algebra L(X; µ) of all (classes of) measurable
functions on X [73, Chapter 5, Proposition 5.3.2].
Note that on U there is no reasonable topology anymore. So U does not fit into
the usual framework of operator algebras. The following tells us that we have
gained good ring theoretical properties.
Proposition 2.4. The algebra U is a von Neumann regular ring, i.e. for every
a ∈ U there exists b ∈ U so that aba = a.
Proof. Every a ∈ U has a unique polar decomposition a = us where u is a partial
isometry in A and s ∈ U is nonnegative and selfadjoint. We have u∗ us = s,
compare Proposition 11.5 and Proposition 11.9. So if we can solve the equation
sts = s with some t ∈ U we are done because with b = tu∗ we get
aba = ustu∗ us = usts = us = a.
Define a function f on the real half line by
0 if λ = 0
.
f (λ) =
1
if λ > 0
λ
The function calculus 11.4 tells us that sf (s)s = s.
This simple fact has a number of consequences. In Appendix II we collect
several alternative definitions of von Neumann regularity and a few of the most
important properties. From homological algebra’s point of view von Neumann
regular rings constitute a distinguished class of rings. They are exactly the rings
with vanishing weak- (or Tor-) dimension.
Note 2.5. A ring R is von Neumann regular if and only if every module is flat,
i.e. for every R-module M the functor − ⊗R M is exact.
Proof. Compare 12.1.
6
Passing from A to U more operators become invertible since unbounded inverses
are allowed.
Lemma 2.6. Let a ∈ U be an operator. The following statements are equivalent.
(i) a is invertible in U.
(ii) a is injective as an operator, i.e. ker(a : dom(a) → H) = 0.
(iii) a has dense image, i.e. im(a) = a(dom(a)) = H.
(iv) la : U → U given by b 7→ ab is an isomorphism of right U-modules.
If moreover a ∈ A ⊂ U the above statements are also equivalent to:
(v) a is a non-zerodivisor in A.
(vi) la : A → A, b 7→ ab is injective.
Proof. (i) ⇔ (ii) ⇔ (iii): We only show (ii) ⇒ (i) and (iii) ⇒ (i). Let a = us be
the polar decomposition of a. Here u is a partial isometry and hence p = u∗ u
and q = uu∗ are projections. We have
im(a) = im(u) = im(uu∗ ) = im(q),
ker(a) = ker(u) = ker(u∗ u) = ker(p).
and
Now if im(a) = H, then q = id and 0 = trA (id) − trA (q) = trA (id) − trA (p) =
trA (id − p). Since the trace is faithful this implies p = id and therefore u is a
unitary operator and in particular invertible. In the case ker(a) = 0 we argue
similarly. Now since ker(s) = ker(u) = 0 we can use the function calculus
(Proposition 11.4) to define an inverse f (s)u∗ with f (λ) = λ1 for λ 6= 0.
(i) ⇔ (iv) is clear.
(i) ⇒ (v) and (v) ⇒ (vi) are easy. It remains to show (vi) ⇒ (ii). Suppose the
bounded operator a ∈ A has a nontrivial kernel. If pker(a) denotes the projection
onto the kernel we have apker(a) = 0 and we see that la is not injective.
An operator with these properties is also called a weak isomorphism. The
function calculus also allows us to write every operator as a fraction.
Lemma 2.7. Every b ∈ U can be written in the form
b = at−1
with a, t ∈ A and t invertible in U.
Proof. Let b = us be the polar decomposition of b (compare Proposition 11.5).
We want to replace s by an invertible operator and use the function calculus.
Let p be the projection 1 − u∗ u. Since uu∗ u = u we have b = us = u(p + s).
Since ker(s) = ker(us) = ker(u∗ u) the projection onto the kernel of s is p. In
7
particular s0 = p + s is a selfadjoint positive operator with trivial kernel. Now
take for instance
1
λ≤1
λ
λ≤1
f (λ) =
.
and g(λ) =
1
λ
>1
1
λ>1
λ
Then f (s0 ) and g(s0 ) are bounded operators affiliated to A and
us = us0 = uf (s0 )g(s0 )−1
with uf (s0 ), g(s0 ) ∈ A and g(s) invertible in U.
Now we show that U is an Ore localization of A. For the relevant definitions we
refer to the first subsection of Appendix III.
Proposition 2.8. An operator in A is a non-zerodivisor if and only if it is
invertible in U. The ring A satisfies both Ore conditions with respect to the set
T of all non-zerodivisors and the (classical) ring of fractions AT −1 is isomorphic
to the algebra of affiliated operators U.
Proof. The first statement was already shown in Lemma 2.6. Now suppose
(a, t) ∈ A × T is given. By the preceding Lemma we can write t−1 a ∈ U as
−1
a 0 t0
with (a0 , t0 ) ∈ A × T . This implies the right Ore condition for the pair
(A, T ). Applying the anti-isomorphism ∗ : U → U yields the right handed
version. Now since the inclusion A ⊂ U is T -inverting, the universal property
of AT −1 gives us a map AT −1 → U. Using again Lemma 2.7 one verifies that
this map is an isomorphism, compare Proposition 13.17(ii).
Corollary 2.9. The ring U is flat over A, i.e. the functor − ⊗A U is exact.
Proof. Ore localization is an exact functor. Compare 13.6.
In the next section we will need the following strengthening of von Neumann
regularity.
Proposition 2.10. The algebra U of operators affiliated to a finite von Neumann algebra is a unit regular ring, i.e. for every element a ∈ U there exists a
unit b ∈ U × with aba = a.
Proof. We have to modify the proof given for von Neumann regularity above
(2.4). Let a = us be the polar decomposition. Let p be the projection uu∗
and q = u∗ u. By definition, this means that p and q are Murray von Neumann
equivalent (p ∼MvN q). Note that p, q, u and u∗ are bounded and therefore
in A. In a finite von Neumann algebra p ∼MvN q implies 1 − p ∼MvN 1 − q.
Compare [82, Chapter V, Proposition 1.38]. (This follows also because one
knows that p ∼MvN q holds if and only if trZ(A) (p) = trZ(A) (q), where trZ(A)
denotes the center valued trace [41, Proposition 8.4.8 on page 532].) Therefore
there exists a partial isometry v with 1 − p = vv ∗ and 1 − q = v ∗ v. Now u∗ + v ∗
is an isometry and in particular invertible since
(u∗ + v ∗ )(u + v) = u∗ u + v ∗ u + u∗ v + v ∗ v = q + 1 − q = 1
8
and similar for (u + v)(u∗ + v ∗ ). Here we used the orthogonal decomposition
H = pH ⊕ (1 − p)H = im u ⊕ ker v ∗ = ker u∗ ⊕ im v to check that v ∗ u = u∗ v = 0.
Moreover we have (u∗ + v ∗ )u = u∗ u and we get with b = t(u∗ + v ∗ )
aba = ust(u∗ + v ∗ )us = usts.
It remains to be found an invertible t ∈ U with sts = s. So take
1 if λ = 0
.
f (λ) =
1
if λ > 0
λ
The function calculus implies sf (s)s = s.
Another useful refinement of von Neumann regularity which is very natural in
the context of operator algebras is ∗-regularity.
Note 2.11. The ∗-algebra U is a ∗-regular ring, i.e. it is a von Neumann regular
∗-ring in which a∗ a = 0 implies a = 0.
Proof. If a is densely defined and closed the domain of a∗ a is a common core
for a and a∗ a, i.e. a can be reconstructed (as the closure) from its restriction
to dom(a∗ a). Therefore 0 =< a∗ a(x), x >=< a(x), a(x) >= |a(x)|2 for all
x ∈ dom(a∗ a) implies a = 0. Compare [73, Theorem 5.1.9].
For more on unit regular and ∗-regular rings see Appendix II.
9
3
Dimensions
The main aim of this section is to prove that there is a well-behaved notion of
dimension for arbitrary U-modules (Theorem 3.12), and that − ⊗A U induces
an isomorphism in K0 (Theorem 3.8). As in the preceding section A denotes a
finite von Neumann algebra and U the algebra of operators affiliated to A.
3.1
A-modules, U-modules and Hilbert A-modules
On our way to a dimension for arbitrary modules we start with finitely generated
ideals of the ring U itself, pass to finitely generated projective modules, then to
finitely generated modules and finally to arbitrary modules. In this subsection
we mainly deal with the first step. But it is also preparatory for the passage to
finitely generated modules. The results for finitely generated projective modules
can also be formulated in terms of A-modules or of so called Hilbert A-modules.
To clarify the relation between these different approaches we systematically deal
with all of them.
It is convenient to organize all submodules of a given finitely generated projective
module in one object: The lattice of submodules.
A lattice is a partially ordered set (L, ≤) in which any two elements {x, y} have
a least upper bound and a greatest lower bound. A lattice is called complete if
for every subset S ⊂ L there exists the least upper bound denoted by sup(S) ∈ L
and the greatest lower bound inf(S) ∈ L. Let L and L0 be complete lattices.
Suppose f : L → L0 is an order isomorphism of partially ordered sets, then it
is also a lattice isomorphism in the sense that f (sup(S)) = sup(f (S)) and
f (inf(S)) = inf(f (S)).
Before we give some examples let us recall the definition of a Hilbert A-module:
The left regular representation for group von Neumann algebras has an analogue
for an arbitrary finite von Neumann algebra. Let l2 (A) = l2 (A; tr) be the Hilbert
space completion of A with respect to the inner product given by
< a, b >= tr(b∗ a).
(This is known as the GNS construction.). There are natural left and right
A-module structures on l2 (A) which give rise to embeddings
A → B(l2 (A))
and
Aop → B(l2 (A)).
It turns out that with respect to these embeddings A0 = Aop and (Aop )0 = A.
We will always consider A as a subalgebra of B(l2 (A)). So for us l2 (A) is the
defining Hilbert space for A.
Definition 3.1. A finitely generated Hilbert A-module is a Hilbert space H
together with a continuous right A-module structure such that there exists a
right A-linear isometric embedding of H onto a closed subspace of l2 (A)n for
some n ∈ N. A morphism of Hilbert A-modules is a bounded right A-linear
map.
10
Here is now the list of examples of lattices:
(i) The lattice LP roj (A) of projections in a von Neumann algebra A
with order given by p ≤ q iff qp = p. This partial order coincides with the
usual order on positive operators ([62, Theorem 2.3.2]). It is well known
that this lattice is complete ([82, Chapter V, Poposition 1.1])
(ii) For an arbitrary ∗-ring R the set of projections is only a partially ordered
set.
(iii) Let M be a finitely generated Hilbert-A-module. Then the set of closed
Hilbert A-submodules ordered by inclusion forms a lattice LHilb (M ).
If M = l2 (A) is the defining Hilbert space for the von Neumann algebra
A, then this lattice is known as the lattice of Hilbert subspaces affiliated
to A, i.e K ⊂ l2 (A) is affiliated if the corresponding projection pK belongs
to A (operating from the left).
(iv) Given an arbitrary ring R and a right module MR one can consider the lattice of all submodules Lall (MR ). This lattice is complete. Supremum
and infimum correspond to sum respectively intersection of modules.
(v) Let R be a coherent ring (compare 14.5) and let PR a finitely generated projective right R-module. Then the set Lf g (PR ) of all finitely
generated submodules forms a sublattice of the lattice Lall (PR ). So
sup{M, N } and inf{M, N } again correspond to sum and intersection of
modules. The important point here is, that over a coherent ring the intersection of two such finitely generated submodules is again finitely generated, compare [81, Chapter I, Proposition 13.3]. In particular, all this
applies to a von Neumann regular ring. In that case all finitely generated
submodules are direct summands. If R is a semihereditary ring (e.g. A)
all finitely generated submodules are projective, compare 14.5. Such a
lattice may not be complete. But even if it is complete the two notions
of sup{Mi | i ∈ I} for infinite families of finitely generated submodules
may differ when considered in Lf g (PR ) respectively in Lall (PR ). This
phenomenon occurs in example 3.20.
(vi) If R is a ring and PR a finitely generated projective right R-module, then
the set Lds (PR ) of submodules which are direct summands is a priori
only a partially ordered set. Of course, if R is von Neumann regular, then
Lds (PR ) = Lf g (PR ). But for a semihereditary ring (for example a finite
von Neumann algebra) we have in general only Lds (PR ) ⊂ Lf g (PR ). The
next theorem tells us that Lds (AA ) is order isomorphic to a lattice and
therefore it is a lattice. But note that it is not clear what the operations
sup and inf are in terms of A-modules.
For more information on lattices and further examples see [81, Chapter III].
11
Proposition 3.2. Given a finite von Neumann algebra A and its algebra of
affiliated operators U, all partially ordered sets in the following commutative diagram are complete lattices, and all maps are order isomorphisms and therefore
lattice isomorphisms.
LHilb (l2 (A)) LP roj (A) - Lds (AA )
?
?
LP roj (U) - Lf g (UU )
The maps are given as follows:
LP roj (A) → LHilb (l2 (A))
LP roj (A) → LP roj (U)
LP roj (A) → Lds (AA )
LP roj (A) → Lf g (UU )
Lds (AA ) → Lf g (UU )
p 7→ im(p : l2 (A) → l2 (A))
p 7→ p
p 7→ pA
p 7→ pU
I 7→ IU.
Here IU is the right U-module generated by I in U.
Proof. Commutativity of the diagram is obvious. That all lattices are complete
follows once we have proven that they are all isomorphic from the completeness
of LP roj (A). In order to prove that a map is a lattice isomorphism it is sufficient
to show that it is an order isomorphism. That sending p to im(p : l2 (A) → l2 (A))
yields a bijection LP roj (A) → LHilb (l2 (A)) is well known. One verifies that this
bijection is an order isomorphism. By commutativity of the square it is sufficient
to deal only with three maps in the square. Since projections in U are bounded
operators they already lie in A, so the lattices LP roj (A) and LP roj (U) coincide.
Given a finitely generated right ideal IU in the ∗-regular ring U there is a unique
projection p ∈ U such that pU = IU , compare 12.5. This leads to the bijection
LP roj (U) → Lf g (UU ). If p ≤ q, then multiplying pU ⊂ U from the left by q
leads to pU ⊂ qU. Note that in general an order preserving bijection need not
be an order isomorphism. But of course if pU ⊂ qU, then multiplying from the
left by 1 − q yields p ≤ q. It remains to prove that LP roj (A) → Lds (AA ) is
surjective; then injectivity follows from the commutativity of the diagram. So
given a right ideal I in A which is a direct summand, there is an idempotent
e such that eA = I. We have to replace the idempotent by a projection. The
following lemma finishes the proof. That the lemma applies is the content of
11.2.
Lemma 3.3. In a ∗-ring R where every element of the form 1+a∗a is invertible
the following holds: Given an idempotent e there always exists a projection p,
such that pR = eR.
Proof. Set z = 1 − (e∗ − e)2 = 1 + (e∗ − e)∗ (e∗ − e). Then z = z ∗ and ze = ez =
ee∗ e and also z −1 e = ez −1 . If we now set p = ee∗ z −1 then p is a projection and
pe = e and ep = p. This leads to pR = eR.
12
We are not primarily interested in these lattices of submodules, but rather in
the set of isomorphism classes of such modules. This is the first step in passing
from embedded submodules to abstract finitely generated projective modules
and then later to arbitrary modules. The point is that over a unit-regular
ring isomorphism of submodules can be expressed in lattice theoretic terms.
Note that a direct sum decomposition P = M ⊕ N can be characterized by
inf{M, N } = 0 and sup{M, N } = P .
Lemma 3.4. Let R be a unit-regular ring and let RR be the ring considered as
a right R-module. Two finitely generated submodules L and M are isomorphic
if and only if they have a common complement in RR , i.e. a submodule N exists
with RR = M ⊕ N and RR = L ⊕ N .
Proof. Compare Corollary 4.4 and Theorem 4.5 in [30].
We obtain the following refined information on the diagram in Proposition 3.2.
Proposition 3.5. The lattice isomorphisms in Proposition 3.2 induce bijections of isomorphism classes, where isomorphism of projections p ∼
= q means
there exist elements x and y in A respectively U such that p = xy and q = yx.
Proof. Again we only have to deal with four of the five maps. The statement for
the map Lds (AA ) → Lf g (UU ) will follow from the commutativity of the square.
We begin with the maps LP roj (A) → Lds (AA ) and LP roj (U) → Lf g (UU ). Let
R be an arbitrary ring. If p = xy and q = yx, then left multiplication by
x respectively y yield mutually inverse homomorphisms between pR and qR.
On the other hand given such mutually inverse homomorphisms the image of
p respectively q under these homomorphisms are possible choices for x and
y. Next we handle the map LP roj (A) → LP roj (U). The only difficulty is
to show that if p and q are isomorphic (alias algebraically equivalent) inside
U, then they are already isomorphic in A. The converse is obviously true.
From Lemma 3.4 above we know that isomorphic finitely generated ideals in U
have a common complement. We have thus expressed isomorphism in lattice
theoretic terms. Since we already know from Proposition 3.2 that the map is
a lattice isomorphism, p and q have a common complement in LP roj (A). The
following lemma, which is due to Kaplansky, finishes the proof for the map
LP roj (A) → LP roj (U) since partial isometries are bounded and therefore in A.
The lemma also tells us that projections p and q in A are isomorphic or algebraically equivalent if and only if they are Murray von Neumann equivalent.
Remains to be examined the map LP roj (A) → LHilb (l2 (A)). That isomorphic
affiliated subspaces of l2 (A) correspond to Murray von Neumann equivalent
projections was the original motivation for the definition of this notion of equivalence. On the one hand the partial isometry gives an isomorphism between
affiliated subspaces. On the other hand one has to replace an arbitrary isomorphism by an isometry using the polar decomposition.
Lemma 3.6. If two projections p and q in a von Neumann algebra A have a
common complement in the lattice of projections LP roj (A), then they are already
13
Murray von Neumann equivalent, i.e. there is a partial isometry u ∈ A such that
p = u∗ u and q = uu∗ .
Proof. See [42, Theorem 6.6(b)]. There it is proven more generally for AW ∗ algebras.
Since Mn (A) is again a finite von Neumann algebra and its algebra of affiliated
operators is isomorphic to Mn (U) one can apply the above results to matrix
algebras.
Corollary 3.7. There is a commutative diagram of complete lattices and lattice
isomorphisms, where all the maps are compatible with the different notions of
isomorphism for the elements of the lattices.
LHilb (l2 (A)n ) LP roj (Mn (A)) - Lds (Mn (A)Mn (A) ) - Lds (AnA )
?
?
?
LP roj (Mn (U)) - Lf g (Mn (U)Mn (U ) ) - Lf g (UUn )
There are stabilization maps
LHilb (l2 (A)n )
LP roj (Mn (A))
Lf g (UUn )
→
LHilb (l2 (A)n+1 )
→
Lf g (UUn+1 )
→ LP roj (Mn+1 (A))
...
and these maps are compatible with the above lattice isomorphisms and the different notions of isomorphism for the elements of the lattices.
Proof. The map Lds (Mn (A)Mn (A) ) → Lds (AnA ) is given by − ⊗Mn (A) AnA followed by the map induced from the natural isomorphism of right A modules
Mn (A) ⊗Mn (A) AnA ∼
= AnA . Morita equivalence tells us that − ⊗A AnMn (A) followed by a natural isomorphism is an inverse of this map. The same argument
applies to U. The vertical map on the right is given by mapping a submodule
M ⊂ An to the U-module it generates inside U n . Since the diagram commutes,
this map is also a lattice isomorphism.
For a ring R let Proj(R) denote the abelian monoid of isomorphism classes of
finitely generated projective modules. An immediate consequence of the above
is the following.
Theorem 3.8. The functor − ⊗A U induces an isomorphism of monoids
Proj(A) → Proj(U).
In particular the natural map
K0 (A) → K0 (U)
is an isomorphism.
14
Proof. The maps Lds (AnA ) → Lf g (UUn ) are compatible with isomorphism, stabilization and direct sums.
For more on the algebraic K-theory of A and U the reader should consult
Section 10.
3.2
A Notion of Dimension for U-modules
Let us now look at dimension functions. Given a faithful normal trace trA :
A → C we obtain a map
dim : LP roj (A) → R
p 7→ trA (p).
Extending the trace to matrices by
trMn (A) ((ai,j )) =
X
trA (ai,i )
yields maps
dim : LP roj (Mn (A)) → R
which are compatible with the stabilization maps. Of course we normalize the
traces such that trMn (A) (1n ) = n. Because of the trace property dim is welldefined on isomorphism classes of projections. Given a finitely generated projective module over A or U there is always an isomorphic module M which is
a direct summand in An respectively U n for some n ∈ N. Similarly a finitely
generated Hilbert A-module is isomorphic to a closed submodule of l2 (A)n for
some n ∈ N. Sending these modules through the diagram of 3.7 to their corresponding projections in Mn (A) and taking the trace gives a real number: The
dimension of M . Of course for Hilbert A-modules this is the dimension considered for example in [14], [10], [11]. For an overview of possible applications
see [54]. For A-modules we get the notion of dimension considered in [51]. The
following proposition formulates the corresponding result for finitely generated
projective U-modules.
Proposition 3.9. There is a well-defined additive real valued notion of dimension for finitely generated projective U-modules. More precisely: Given a finitely
generated projective U-module M we can assign to it a real number dimU (M ),
such that:
(i) dimU (M ) depends only on the isomorphism class of M .
(ii) dimU (M ⊕ N ) = dimU (M ) + dimU (N ).
(iii) dimU (M ⊗A U) = dimA (M ) if M is a finitely generated projective Amodule.
(iv) M = 0 if and only if dimU (M ) = 0.
15
Proof. (i) and (iii) follow immediately from 3.7. Up to isomorphism and stabilization a direct sum of modules corresponds to the block diagonal sum of
projections, this yields (ii). Faithfulness of the trace implies (iv).
So far we have not used the fact that the lattices are complete. We will see that
this will enable us to extend the notion of dimension to arbitrary U-modules.
The following definition is completely analoguous to the definition of the dimension for A-modules given in [52].
Definition 3.10. Let M be an arbitrary U-module. Define dim0U (M ) ∈ [0, ∞]
as
dim0U (M ) = sup{dimU (P ) | P ⊂ M, P fin. gen. projective submodule}.
The next lemma is the main technical point in proving that this dimension is
well-behaved and of course it uses the completeness of the lattices.
If K is a submodule of the finitely generated projective module M we define
\
K=
Q ⊂ M,
K⊂Q⊂M
where the intersection is over all finitely generated submodules Q of M , which
contain K.
Lemma 3.11. Let K be a submodule of U n . Since the lattice Lf g (UUn ) is complete the supremum of the set {P | P ⊂ K, P finitely generated} exists.
(i) We have
K = sup{P | P ⊂ K, P finitely generated}
and this module is finitely generated and therefore projective.
(ii) We have
dim0U (K) = dimU (K).
Proof. (i) Let {Pi | i ∈ I} be the system of finitely generated submodules and
{Qj | j ∈ J} be the system of finitely generated modules containing K. Since
every element of K generates a finitely generated submodule of K we know
that K ⊂ sup{Pi | i ∈ I}. Since the lattice is complete sup{Pi | i ∈ I} is one
of the finitely generated modules containing K in the definition of K. We
get K ⊂ sup{Pi | i ∈ I}. Since Pi ⊂ Qj for i, j arbitrary it follows that
sup{Pi | i ∈ I} ⊂ Qj for all j ∈ J and therefore sup{Pi | i ∈ I} ⊂ K.
(ii) From (i) we know that K is finitely generated projective. Let p be the
projection corresponding to K and pi be those corresponding to the Pi , then
p is the limit of the increasing net pi and normality of the trace implies the
result.
16
Theorem 3.12. Let dimA be the dimension for A-modules considered in [52].
The dimension dim0U defined above has the following properties.
(i) Invariance under isomorphisms: dim0U (M ) depends only on the isomorphism class of M .
(ii) Extension: If Q is a finitely generated projective module, then
dim0U (Q) = dimU (Q).
(iii) Additivity: Given an exact sequence of modules
0 → M0 → M1 → M2 → 0
we have
dim0U (M1 ) = dim0U (M0 ) + dim0U (M2 ).
S
(iv) Cofinality: Let M = i∈I Mi be a directed union of submodules (i.e.given
i, j ∈ I there always exists an k ∈ I, such that Mi , Mj ⊂ Mk ) then
dim0U (M ) = sup{dim0U (Mi ) | i ∈ I}.
These four properties determine dim0U uniquely. Moreover the following holds.
(v) If M is an A-module, then dim0U (M ⊗A U) = dimA (M ).
(vi) If M is finitely generated projective, then dim0U (M ) = 0 if and only if
M = 0.
(vii) Monotony: M ⊂ N implies dim0U (M ) ≤ dim0U (N ).
Notation 3.13. After having established the proof we will write dimU instead
of dim0U . This is justified by (ii).
Proof. (i) Invariance under isomorphisms: This follows from the definition and
the corresponding property 3.9 (i) for finitely generated projective modules.
(ii) Extension property: Let P be a finitely generated projective submodule of
the finitely generated projective module Q, then since U is von Neumann regular
P is a direct summand of Q. The additivity for finitely generated projective
modules 3.9 (ii) implies that
S dimU (P ) ≤ dimU (Q). The claim follows.
(iv) Cofinality: Let M = i∈I Mi be a directed union. It is obvious from the
definition that dim0U (Mi ) ≤ dim0U (M ) and therefore sup{dim0U (Mi ) | i ∈ I} ≤
dim0U (M ). Let now P ⊂ M be finitely generated projective. Since the system is
directed there is an i ∈ I such that Mi contains all generators of P and therefore
P itself. It follows that dimU (P ) ≤ dim0U (Mi ) ≤ sup{dim0U (Mi ) | i ∈ I} and
finally
sup{dimU (P ) | P ⊂ M, P fin. gen. projective} ≤ sup{dim0U (Mi ) | i ∈ I}.
17
(vii) Monotony: M ⊂ N implies dim0U (M ) ≤ dim0U (N ), since on the left one
has to take the supremum over a smaller set of numbers.
(iii) Additivity: Let
0 → M0
- M1
i
- M2 → 0
p
be an exact sequence. For every finitely generated projective submodule P ⊂ M2
there is an induced sequence
0 → M0 → p−1 (P ) → P → 0
which splits. So p−1 (P ) ∼
= M0 ⊕ P . From
{Q ⊕ P | Q ⊂ M0 , Q f.g.proj.} ⊂ {Q0 | Q0 ⊂ M0 ⊕ P, Q0 f.g.proj. }
it follows that dim0U (M0 ) + dimU (P ) ≤ dim0U (M0 ⊕ P ). The monotony implies
dim0U (M0 ) + dimU (P ) ≤ dim0U (p−1 (P )) ≤ dim0U (M1 ). Taking the supremum
over all P leads to dim0U (M0 ) + dim0U (M2 ) ≤ dim0U (M1 ). Now to the reverse
inequality: Let Q ⊂ M1 be a finitely generated projective submodule. There
are two exact sequences:
0 → i(M0 ) ∩ Q → Q → p(Q) → 0
0 → i(M0 ) ∩ Q → Q → Q/i(M0 ) ∩ Q → 0
Because of 3.11 (i) the second one is an exact sequence of finitely generated
projective modules and hence by 3.9 we already know that
dim0U (Q) = dim0U (i(M0 ) ∩ Q) + dim0U (Q/i(M0 ) ∩ Q).
Using 3.11 (ii)
dim0U (Q) = dim0U (i(M0 ) ∩ Q) + dim0U (Q/i(M0 ) ∩ Q)
follows. Now there is a split epimorphism p(Q) → Q/i(M0) ∩ Q yielding dim0U (p(Q)) ≥
dim0U (Q/i(M0 ) ∩ Q) by monotony. So
dim0U (Q) ≤ dim0U (i(M0 ) ∩ Q)) + dim0U (p(Q))
and monotony gives
dim0U (Q) ≤ dim0U (M0 ) + dim0U (M2 ).
Taking the supremum over all finitely generated submodules Q ⊂ M1 finally
gives dim0U (M1 ) ≤ dim0U (M0 ) + dim0U (M2 ).
(vi) Was already proven in 3.9.
Let us now prove uniqueness. This is done in several steps.
Step1: The extension property determines dim0U uniquely on finitely generated
projective modules.
18
Step2: Let now K ⊂ Q be a submodule of a finitely generated projective module.
The
S module K is the directed union of its finitely generated submodules K =
i∈I Ki . Since U is von Neumann regular the Ki are projective (semihereditary
would be sufficient here). Now dim0U (K) is uniquely determined by cofinality
and Step1.
Step3: If M is finitely generated there is an exact sequence 0 → K → U n →
M → 0. Additivity together with Step2 implies the result for finitely generated
modules.
Step4: An arbitrary module is the directed union of its finitely generated submodules and again one applies cofinality.
(v) The proof follows the same pattern as the proof of uniqueness. Note that it
is shown in [52] that the dimension dimA for A-modules also has the properties
(i) to (iv).
Step1: For finitely generated projective A-modules this is the content of 3.9(iii).
Step2: A submodule K of a finitely generated projective A-module is the directed union of its finitely generated submodules Ki which are projective since
A is semihereditary. Since − ⊗A U is exact and commutes with colimits K ⊗A U
is the directed union of the Ki ⊗A U. Now apply cofinality of dimA and dim0U
and use Step1.
Step3: For a finitely generated module M applying −⊗A U to the exact sequence
0 → K → An → M → 0 yields an exact sequence. Now use Step2 and the
additivity of dimA respectively dim0U .
Step4: An arbitrary module is the directed union of its finitely generated submodules. Proceed as in Step2 and use Step3.
Note that this proof is independent of that one for dimA in [52] which uses properties of the functor ν constructed in [51]. One could therefore take dimA (M ) =
dimU (M ⊗A U) as a definition.
3.3
The Passage from A-modules to U-modules
Thinking of U as a localization of A it is no surprise that on the one hand
we lose information by passing to U-modules, but on the other hand U-modules
have better properties. For example, every finitely presented U-module is finitely
generated projective and the category of finitely generated projective U-modules
is abelian. We will now investigate this passage systematically.
Definition 3.14. For an A-module M we define its torsion submodule tM as
tM = ker(M → M ⊗A U).
A module M is called a torsion module if M ⊗A U = 0 or equivalently tM =
M . A module is called torsionfree if tM = 0.
This is consistent with the terminology for example in [81, page 57] because U
is isomorphic to the classical ring of fractions of A. An element m ∈ M lies in
19
tM if and only if it is a torsion element in the following sense: There exists a
non-zerodivisor s ∈ A, such that ms = 0. Compare [81, Chapter II, Corollary
3.3]. The module M/tM is torsionfree since ms = ms = 0 ∈ M/tM implies
the existence of s0 with mss0 = 0 and therefore m ∈ tM .
On the other hand, following [52, page 146] we make the following definition.
Definition 3.15. Let M be an A-module, then
[
TM =
N,
where the union is over all N ⊂ M with dimA (N ) = dimU (N ⊗A U) = 0. We
denote by PM the cokernel of the inclusion TM ⊂ M .
This is indeed a submodule because for two submodules N , N 0 ⊂ M with
dimA (N ) = dimA (N 0 ) = 0 the additivity of the dimension together with
N + N 0 /N ∼
= N 0 /N 0 ∩ N
implies dimA (N + N 0 ) = 0. Note that dimA (TM ) = 0 by cofinality and TM is
the largest submodule with vanishing dimension.
Note 3.16. Both t and T are left exact functors.
Proof. Under a module homomorphism torsion elements are mapped to torsion elements and a homomorphic image of a zero-dimensional module is zerodimensional by additivity. If L ⊂ M is a submodule, then tL = L ∩ tM and
TL = L ∩ TM since L ∩ TM has vanishing dimension as a submodule of TM .
This implies left exactness.
We will now investigate the relation between tM and TM and the question to
what degree the U-dimension is faithful.
Finitely generated projective modules. If we restrict ourselves to finitely
generated projective modules, we have seen in 3.8 that isomorphism classes of
A- and U-modules are in bijective correspondence via − ⊗A U, and by adding a
suitable complement one verifies that the natural map M → M ⊗A U is injective.
A finitely generated projective module over A or U is trivial if and only if its
dimension vanishes by 3.9. Therefore tM = TM = 0 for finitely generated
projective A-modules.
Finitely presented modules. Let us now consider finitely presented Amodules. This category of modules was investigated in [51]. Since the ring
A is semihereditary it is an abelian category. (This is more generally true iff
the ring is coherent.) Recall that Proj(R) denotes the monoid of isomorphism
classes of finitely generated projective modules over R.
Proposition 3.17. Let M be a finitely presented A-module. Then
(i) M ⊗A U is finitely generated projective.
(ii) TM = tM .
20
(iii) M is a torsion module if and only if dimA (M ) = 0. Every torsion module
is the cokernel of a weak isomorphism.
(iv) PM = M/TM is projective and M ∼
= PM ⊕ TM .
(v) Under the isomorphism Proj(A) → Proj(U), respectively K0 (A) → K0 (U)
the class [PM ] corresponds to [M ⊗A U].
Proof. (i) By right exactness of the tensor product M ⊗A U is finitely presented.
Over the von Neumann regular ring U this implies being finitely generated
projective. (iv) is proven in [51]. Now M ⊗A U ∼
= PM ⊗A U ⊕ TM ⊗A U ∼
= PM
∼
because TM = coker(f ) for some weak isomorphism f : An → An and by 2.6
and right exactness of the tensor product we get TM ⊗A U ∼
= coker(f ) ⊗A U ∼
=
∼
coker(f ⊗A idU ) = 0. The rest follows.
Arbitrary modules. In general tM and TM differ. Counterexamples can
already be realized by finitely generated modules. More precisely the following
holds.
Proposition 3.18. Let M be an A-module.
(i) tM ⊂ TM and torsion modules have vanishing dimensions.
(ii) There exists a finitely generated A-module with tM = 0 and TM = M .
(iii) M is a torsion module if and only if it is the directed union of quotients
of finitely presented torsion modules.
(iv) If M is finitely generated PM = M/TM is projective and M ∼
= PM ⊕
TM .
(v) There exists a finitely generated module M with
dimA (M ) = dimU (M ⊗A U) = 0
but M ⊗A U =
6 0.
Proof. (i) It suffices to show that dimA (tM ) = dimU (tM ⊗A U) = 0. Now
tM consists of torsion elements and therefore tM ⊗A U = 0. (v) follows from
(ii) and we will give an example of such a module below. (iv) was proven in
[52]. Let us prove (iii): A quotient of a torsion module is a torsion module, and
a directed union of torsion modules is again S
a torsion module. On the other
hand suppose M is a torsion module. M = i∈I Mi is the directed union of
its finitely generated submodules. Since any submodule of a torsion module
is a torsion module it remains to be shown that a finitely generated torsion
module N is always a quotient of a finitely presented torsion module. Choose a
surjection p : An → N and let K = ker(p) be the kernel. Since N ∼
= An /K is a
n
torsion module, for every a ∈ A there exists a non-zerodivisor s ∈ A such that
as ∈ K. Let ei = (0, . . . , 1, . . . , 0)tr be the standard basis for An and choose si
with ei si ∈ K. Note that
diag(0, . . . , si , . . . , 0)ei = ei si .
21
Here diag(b1 , . . . , bn ) denotes the diagonal matrix with the corresponding entries. Since K is a right A-module we have for an arbitrary vector (a1 , . . . , an )tr ∈
An that
X
diag(s1 , . . . , sn )(a1 , . . . , an )tr =
ei si ai ∈ K.
Let S denote the right linear map An → An corresponding to the diagonal
matrix diag(s1 , . . . , sn ), then im(S) ⊂ K and N ∼
a quotient of
= An /K is L
n
the finitely presented module A /im(S). Moreover An /im(S) ∼
A/si A is a
=
torsion module by Proposition 3.17(iii) and 2.6.
Note 3.19. In [55, Definition 2.1] we defined a module to be cofinal-measurable
if all its finitely generated submodules are quotients of finitely presented zerodimensional modules. By Proposition 3.17(iii) and 3.18(iii) we see that a module
is cofinal-measurable if and only if it is a torsion module.
Example 3.20. We will now give two counterexamples to finish the proof of
the above proposition. Both examples follow the same pattern. Let Ii , i ∈ I
be a directed family of right ideals in A such that each Ii S
is a direct summand,
dimA (Ii ) < 1 and supi∈I (dimA (Ii )) = 1. Note that I = i∈I Ii 6= A, because
1 ∈ Ii for some i would contradict dimA (Ii ) < 1. Since Ii is a direct summand
A/Ii → (A/Ii ) ⊗A U is injective. Using that − ⊗A U is exact and commutes
with colimits one verifies that
[
A/I → (A/I) ⊗A U ∼
= A ⊗A U/I ⊗A U ∼
= A ⊗A U/ (Ii ⊗A U)
is injective as well. Therefore t(A/I) = 0. On the other hand additivity and
cofinality of the dimension imply dimA (A/I) = dimU ((A/I) ⊗A U) = 0. Here
are two concrete examples where such a situation arises.
(i) Take A = L∞ (S1 , µ), the essentially bounded functions on the unit circle
with respect to the normalized Haar measure µ on S1 . Let Xi , i ∈ N be
an increasing sequence of measurable subsets of S1 such that µ(Xi ) < 1
and supi∈N (µ(Xi )) = 1. The corresponding characteristic functions χXi
generate ideals in A with the desired properties, since dimA (χXi A) =
µ(Xi ).
(ii) Let Γ be an infinite locally finite group, i.e. every finitely generated subgroup is finite. Take A as N Γ. Let I denote the set of finiteP
subgroups.
P For
every H ∈ I let H : CH → C denote the augmentation
ah h 7→
ah .
Since H is finite the augmentation ideal ker(h ) is a direct summand of
CH. Since CH = N H and − ⊗N H N Γ is exact and compatible with
dimensions we get that ker(H ) ⊗CH N Γ is a direct summand of N Γ and
dimN Γ (ker(H ) ⊗CH N Γ) = 1 −
1
.
|H|
The system is directed because for finite subgroups H and G with H ⊂ G
we have ker(H ) ⊗CH CG ⊂ ker(G ). If we tensor the exact sequence
[
ker(H ) ⊗CH CΓ → CΓ → C ⊗CΓ CΓ → 0
H
22
with N Γ and use compatibility with colimits and right exactness we get
[
H0Γ (EΓ; N Γ) ∼
= C ⊗CΓ N Γ ∼
= N Γ/ ker(H ) ⊗CH N Γ.
H
So we can even realize the counterexample as the homology of a space.
23
4
New Interpretation of L2-Invariants
We will now briefly summarize how the notions developed in the last section
apply in order to provide alternative descriptions of L2 -invariants. Recall the
following definitions from [52, Section 4]. Let Γ be a group. Given an arbitrary
Γ-space X one defines the singular homology of X with twisted coefficients in
the ZΓ-module N Γ as
HpΓ (X; N Γ) = Hp (C∗sing (X) ⊗ZΓ N Γ),
where C∗sing (X) is the singular chain complex of X considered as a complex of
right ZΓ-modules. Here HpΓ (X; N Γ) is still a right N Γ-module and therefore
Γ
b(2)
p (X) = dimA (Hp (X; N Γ))
makes sense. This definition extends the definition of L2 -Betti numbers via
Hilbert N Γ-modules for regular coverings of CW-complexes of finite type to
arbitrary Γ-spaces. Completely analoguous we make the following definition.
Definition 4.1. Let X be a Γ-space, then
HpΓ (X; UΓ) = Hp (C∗sing (X) ⊗ZΓ UΓ).
Proposition 4.2. Let X be an arbitrary Γ-space.
(i) HpΓ (X; N Γ) ⊗N Γ UΓ ∼
= HpΓ (X; UΓ) as UΓ-modules.
(2)
(ii) bp (X) = dimU (HpΓ (X; UΓ)).
(iii) tHpΓ (X; N Γ) = ker(HpΓ (X; N Γ) → HpΓ (X; UΓ)).
If X is a regular covering of a CW-complex of finite type, then
(iv) HpΓ (X; N Γ) is finitely presented, HpΓ (X; UΓ) is finitely generated projective and under the isomorphism
Proj(NΓ) → Proj(UΓ), respectively
K0 (N Γ) → K0 (UΓ) the class PHpΓ (X; N Γ) corresponds to HpΓ (X; UΓ) .
(v) tHpΓ (X; N Γ) = THpΓ (X; N Γ) = ker(HpΓ (X; N Γ) → HpΓ (X; UΓ)).
Proof. Since − ⊗N Γ UΓ is exact it commutes with homology. If X is a regular
covering of finite type the singular chain complex is quasi-isomorphic to the
cellular chain complex which consists of finitely generated free ZΓ-modules.
Since finitely presented N Γ-modules form an abelian category the homology
modules HpΓ (X; N Γ) are again finitely presented. The rest follows from the
preceding results about t, T and the dimension.
As far as dimension is concerned one can therefore work with U-modules. Tensoring with U is the algebraic analogue of the passage from unreduced to reduced
L2 -homology in the Hilbert space set-up. If one is interested in finer invariants
24
like the Novikov-Shubin invariants the passage to U-modules is too harsh. The
torsion submodule tHpΓ (X; N Γ) carries the information of the Novikov-Shubin
invariants in case X is a regular covering of a CW-complex of finite type and
seems to be the right candidate to carry similar information in general. Compare
[55] and Note 3.19.
25
5
The Atiyah Conjecture
5.1
The Atiyah Conjecture
As already mentioned in the introduction one of the main motivations to study
the algebras CΓ, DΓ, RΓ, N Γ and UΓ is the following conjecture, which goes
back to a question of Atiyah in [1]. We denote by F in(Γ) the set of finite
1
Z is the additive subgroup of R generated by the set
subgroups of Γ and |F inΓ|
1
1
of rational numbers { |H| |H a finite subgroup of Γ}. In particular |F inΓ|
Z=Z
if Γ is a torsionfree group and if there is a bound on the order of finite subgroups
1
1
|F inΓ| Z = l Z, where l is the least common multiple of these orders. In general
1
|F inΓ| Z ⊂ Q.
Conjecture 5.1 (Atiyah Conjecture). If M is a closed manifold with fundamental group Γ and universal covering M̃ , then
b(2)
p (M̃ ) ∈
1
Z
|FinΓ|
for all p ≥ 0.
There is a topological and a purely algebraic reformulation of this conjecture.
Proposition 5.2. Let Γ be a finitely presented group.The following statements
are equivalent.
(i) For every finitely presented ZΓ-module M
dimU Γ (M ⊗ZΓ UΓ) ∈
1
Z.
|FinΓ|
(ii) For every ZΓ-linear map f between finitely generated free ZΓ-modules
dimU Γ (ker(f ⊗ZΓ idU Γ )) ∈
1
Z.
|FinΓ|
(iii) If X is a connected CW-complex of finite type with fundamental group Γ,
then
b(2)
p (X̃) ∈
1
Z
|FinΓ|
for all p ≥ 0.
(iv) If M is a closed manifold with fundamental group Γ, then
b(2)
p (M̃ ) ∈
1
Z
|FinΓ|
for all p ≥ 0.
Proof. Using the additivity of the dimension we verify that in the statement
of (ii) one can replace the kernel of the map f ⊗ZΓ idU Γ by the image or the
cokernel. Since tensoring is right exact coker(f ⊗ZΓ idU Γ ) ∼
= coker(f )⊗ZΓ UΓ and
we see that (i) and (ii) are equivalent. Let us now prove that (ii) implies (iii).
26
The cellular chain complex (C∗ , d∗ ) = (C∗cell (X̃), d∗ ) of X̃ is a complex of finitely
(2)
generated free right ZΓ-modules. By definition bp (X̃) = dimU Γ Hp (C∗ ⊗ZΓ UΓ).
The exact sequences
0 → ker(dn ⊗ idU Γ ) →
Cn ⊗ UΓ
0 → im(dn+1 ⊗ idU Γ ) → ker(dn ⊗ idU Γ )
→ im(dn ⊗ idU Γ ) → 0
→ Hn (C∗ ⊗ UΓ) → 0
together with (ii) and additivity of dimension imply (iii). Since (iv) is a special
case of (iii) it remains to be shown that (iv) implies (ii). Since Γ is finitely
presented there is a finite 2-complex with fundamental group Γ. Now attach
m 3-cells with a trivial attaching map and n 4-cells in such a way that the
differential d4 realizes f . Embed this finite CW-complex X in RN and take
M as the boundary of a regular neighbourhood. There is a 5-connected map
M → X and therefore H4 (M̃ , UΓ) = H4 (X̃, UΓ) = ker(f ⊗ idU Γ ).
It is convenient to formulate the algebraic version of the Atiyah conjecture not
only for the integral group ring but also for group rings with other coefficients.
Conjecture 5.3 (Atiyah Conjecture with Coefficients). Let R ⊂ C be a
subring of the complex numbers. For every finitely presented RΓ-module M
dimU Γ (M ⊗RΓ UΓ) ∈
1
Z.
|FinΓ|
The conjecture with coefficients R ⊂ C implies the conjecture with integer
coefficients. In fact the results obtained by Linnell, which we will discuss later,
all hold even for complex coefficients. Conversely we have at least:
Lemma 5.4. Let R ⊂ C be a subring and denote by Q(R) the subfield of C
generated by R. The Atiyah conjecture with R-coefficients implies the Atiyah
conjecture with Q(R)-coefficients. In particular the genuine Atiyah conjecture
implies the Atiyah conjecture with Q-coefficients.
Proof. The field Q(R) is the localization RS −1 , where S = R − {0}. Given a
map f : RΓn → RΓm we interpret it as a matrix and multiply by a suitable
s ∈ S to clear denominators. This does not change the dimension of the kernel
or the image of f ⊗ idU Γ since multiplication by s is an isomorphism.
Another easy observation is that the conjecture is stable under directed unions.
S
Lemma 5.5. Suppose Γ = i∈I Γi is the directed union of the subgroups Γi and
the Atiyah conjecture with R-coefficients holds for each Γi , then it also holds for
Γ.
Proof. Every entry of a matrix representing f : RΓn → RΓm involves only
finitely many group elements. Since the union is directed we can find an i ∈ I
such that f is a matrix over RΓi . Now
dimU Γ (coker(f ⊗ idU Γ )) = dimU Γ (coker(f ⊗ idU Γi )) ⊗U Γi UΓ
1
1
= dimU Γi (coker(f ⊗ idU Γi )) ∈
Z⊂
Z
|FinΓi |
|FinΓ|
27
since the dimension behaves well with respect to induction and tensoring is right
exact.
Similarly, at least for torsionfree groups, the conjecture is stable under passage
to subgroups.
The Atiyah conjecture is related to the zero divisor conjecture.
Conjecture 5.6 (Zero Divisor Conjecture). The rational group ring QΓ of
a torsion free group contains no zero divisors.
The converse of this statement
is true, since given a finite subgroup H ⊂ Γ
P
1
the norm element |H|
h
is
an idempotent and therefore a zerodivisor. Of
h∈H
course this conjecture could also be stated with other coefficients.
Note 5.7. The Atiyah conjecture implies the zero divisor conjecture.
Proof. Suppose a ∈ QΓ is a zerodivisor. Left multiplication with a is a ZΓlinear map and therefore dimU Γ ker(a ⊗ idU Γ ) ∈ Z by the Atiyah conjecture with
rational coefficients. Since ker(a ⊗ idU Γ ) is a submodule of ZΓ ⊗ZΓ UΓ ∼
= UΓ, we
have that dimU Γ (ker(a ⊗ idU Γ )) is either 0 or 1. In the second case a = 0. In the
first case ker a ⊂ ker(a ⊗ idU Γ ) = 0 because ker(a ⊗ idU Γ ) is a finitely generated
projective module and such a module vanishes if and only if its dimension is
zero.
We will briefly discuss how the Atiyah conjecture is related to questions about
Euler characteristics of groups. If a group Γ has a finite model for its classifying
space BΓ, then its Euler characteristic χ(Γ) = χ(BΓ) is defined. Note that such
a group is necessarily torsionfree. If Γ0 ⊂ Γ is a subgroup of finite index we have
χ(Γ0 ) = [Γ : Γ0 ] · χ(Γ).
This is the fundamental property which allows to extend the definition to not
necessarily torsionfree groups. A group Γ is said to have virtually a finite classifying space if a subgroup Γ0 of finite index admits a finite classifying space. In
that case we define
χvirt (Γ) =
1
χ(Γ0 ).
[Γ : Γ0 ]
This idea goes back to Wall [84]. It is possible to extend the definition further
to groups Γ which are of finite homological type. For the definition of groups
of finite homological type and more details we refer to Capter IX in [8]. The
following Theorem was conjectured by Serre and proven by Brown [7].
Theorem 5.8 (Serre’s Conjecture). If Γ is of finite homological type, then
χvirt (Γ) ∈
1
Z.
|FinΓ|
Proof. See Theorem 9.3 on page 257 in [8] and page 246 in the same book for
the definition of finite homological type.
28
Now if we consider the passage to a subgroup of finite index the L2 -Betti numbers behave like the Euler characteristic. Namely let Y be a Γ-space and Γ0 be
a subgroup of finite index, then
b(2)
p (Y ; Γ) =
1
b(2) (Y ; Γ0 ).
[Γ : Γ0 ] p
Therefore the L2 -Euler characteristic
χ(2) (Γ) =
∞
X
(−1)p b(2)
p (EΓ),
p=0
which is known to coincide with the ordinary Euler characteristic in the case
of a finite complex, coincides with the virtual Euler characteristic if Γ virtually
admits a finite classifying space. For more details we refer to [11] and [53].
Note 5.9. The Atiyah conjecture implies
χ(2) (Γ) ∈
1
Z.
|FinΓ|
Therefore the theorem of Brown above gives further evidence for the Atiyah
conjecture.
5.2
A Strategy for the Proof
The main objective of this section is the following strategy for a proof of the
above mentioned conjectures.
Theorem 5.10. Let Γ be a discrete group. Suppose there is an intermediate
ring CΓ ⊂ SΓ ⊂ UΓ such that
(A) The ring SΓ is von Neumann regular.
(B) The map iΓ induced by the natural induction maps
M
iΓ :
K0 (CK) → K0 (SΓ)
K∈F inΓ
is surjective.
Then the Atiyah conjecture with complex coefficients is true for the group Γ.
Proof. Suppose M is a finitely presented CΓ-module. It does not define a class in
K0 (CΓ), but since every finitely presented module over the von Neumann regular
ring SΓ is projective, M ⊗CΓ SΓ defines a class in K0 (SΓ). Now
L the surjectivity
of iΓ implies that there is an element ([NK ] − [LK ])K ∈
K∈F inΓ K0 (CK)
which maps to [M ⊗CΓ SΓ]. Here NK and LK are finitely generated projective
CK-modules. Since the Γ-dimension behaves well with respect to induction,
29
and for finite groups it coincides with the complex dimension divided by the
group order, we have
X
dimU Γ ([NK ⊗ UΓ] − [LK ⊗ UΓ])
dimU Γ (ker(f ⊗ idSΓ ) ⊗ UΓ) =
K
=
X
K
=
dimCK (NK ) − dimCK (LK )
X 1
(dimC (NK ) − dimC (LK )) ,
|K|
K
which is obviously in the subgroup
1
|F inΓ| Z.
Roughly speaking condition (A) says that we have to make our ring large enough
to have good ring theoretical properties (remember that at least UΓ itself is von
Neumann regular). Condition (B) says we cannot make it too big because
otherwise its K-theory becomes too large. Here are some remarks on the proof.
(i) The same proof applies if we replace the set F inΓ of finite subgroups
in condition (B) by any set F (Γ) of subgroups for which the Atiyah
conjecture is already known, because then we have for K ∈ F(Γ) and
1
1
[NK ] ∈ K0 (CK) that dimU K (NK ) ∈ |F inK|
Z ⊂ |F inΓ|
Z. For example
one could take the virtually cyclic subgroups of Γ.
(ii) Of course we can formulate a similar statement for other than complex
coefficients. This might be an advantage for verifying (B), compare the
next section.
(iii) Note that we actually proved that (A) and (B) implies
im(K0 (SΓ) → K0 (UΓ)
- R) =
dim
1
Z.
|FinΓ|
(iv) Working with the kernel instead of the cokernel of a map f between finitely
generated free CΓ-modules one verifies that instead of von Neumann regularity it would be sufficient that the ring SΓ is semihereditary and UΓ
is flat as an SΓ-module. Remember that a ring is semihereditary if every
finitely generated submodule of a projective module is projective. This
would imply that im(f ⊗ idSΓ ) and therefore also ker(f ⊗ idSΓ ) is finitely
generated projective. In fact it would be sufficient to ensure that one
of these modules has a finite resolution by finitely generated projective
modules.
(v) It seems that every reasonable construction of a ring SΓ leads to a ring
which is closed under the ∗-operation. We therefore expect SΓ to be a
∗-closed subring of UΓ.
30
Later, when we discuss Linnell’s results on the Atiyah conjecture, we will actually find an SΓ which is not only von Neumann regular but semisimple. Nevertheless we believe that in general the above should be the right formulation for
the following reasons.
Example 5.11. Let Γ be an infinite locally L
finite group (every finitely
S generated subgroup is finite), as for example Γ0 = n∈N Z/nZ. Then Γ = i∈I Γi is
the directed union of its finite subgroups. Using the elementwise criterion
for
S
von Neumann regularity (compare 12.1(i)) one verifies that CΓ = i∈I CΓi is
von Neumann regular. Also (B) holds. Therefore we conclude from the above
1
Z. This group is not finitely
remark that im(K0 (CΓ) → K0 (UΓ) → R) = |F inΓ|
0
generated (for Γ it is Q). Suppose SΓ is a semisimple intermediate ring, then
K0 (SΓ) is finitely generated and the above map to R factorizes over K0 (SΓ).
A contradiction.
We see that in general we cannot expect a semisimple intermediate ring SΓ. On
1
the other hand, when |F inΓ|
Z is a discrete subgroup of R semisimplicity already
follows from (A) and (B):
Proposition 5.12. Suppose the intermediate ring SΓ fulfills the conditions (A)
and (B) and there is a bound on the orders of finite subgroups of Γ, then SΓ is
semisimple.
Proof. By assumption we know that
im(K0 (SΓ) → K0 (UΓ) → R) =
1
1
Z= Z
|FinΓ|
l
for a positive integer l. We will see that there can be no strictly increasing
chain I1 ⊂ I2 ⊂ . . . ⊂ Ir ⊂ SΓ of right ideals with r > l. With 12.3(iii) this
implies the result. Choose xi ∈ Ii \ Ii−1 and let Ji be the ideal generated by
x1 , . . . xi , then the Ji form a strictly increasing chain of finitely generated ideals
of the same length. Since SΓ is von Neumann regular Jr is a direct summand
of SΓ and Ji−1 is a direct summand of Ji . We see that dimU Γ (Ji−1 ⊗ UΓ)
is strictly smaller than dimU Γ (Ji ⊗ UΓ). Here we are using the fact that for
finitely generated projective SΓ-modules the dimension function dimU Γ (−⊗UΓ)
is faithful. Simply represent a finitely generated projective module P over SΓ
by an idempotent e ∈ M(SΓ) and verify that the same idempotent considered
as a matrix over M(UΓ) represents P ⊗SΓ UΓ. Now since dimU Γ is faithful
dimU Γ (P ⊗ UΓ) = 0 implies that P ⊗ UΓ = 0 and therefore also the idempotent
e is trivial. Now dimU Γ (Ji ⊗ UΓ) ∈ [0, 1] ∩ 1l Z since Ji ⊗ UΓ is a submodule of
UΓ. Therefore the chain can at most have length l.
Similarly we have for torsionfree groups:
Proposition 5.13. If Γ is torsionfree and SΓ is an intermediate ring of CΓ ⊂
UΓ, then conditions (A) and (B) hold for SΓ if and only if SΓ is a skew field.
31
1
Z = Z we can take l = 1 in
Proof. Suppose SΓ fulfills (A) and (B). Since |F inΓ|
the proof above. So we have a semisimple ring which has no nontrivial ideals. On
the other hand a skew field is von Neumann regular and K0 (SΓ) = Z generated
by [SΓ] if SΓ is a skew field and therefore (B) is obviously satisfied.
We have seen in the preceding section that the Atiyah conjecture is stable under
directed unions. It is therefore reassuring that conditions (A) and (B) also show
this behaviour. At the same time the following generalizes Example 5.11.
S
Proposition 5.14. Suppose Γ = i∈I Γi is a directed union and there are intermediate rings SΓi with SΓi ⊂ SΓ
Sj for Γi ⊂ Γj which all fulfill the conditions
(A) and (B), then the ring SΓ = i∈I SΓi also fulfills (A) and (B).
Proof. A directed union of von Neumann regular rings is von Neumann regular
and K-theory is compatible with colimits.
An intermediate ring SΓ is by no means uniquely determined by (A) and (B).
Example 5.15. Let Γ = Z be an infinite group. We know that in this case
we
can identify the rings CΓ and UΓ with the ring of Laurent polynomials C z ±1
and the ring L(S1 ) of (classes of) measurable functions on the unit circle S1 ,
where we embed the circle in the complex plane and z denotes the standard
coordinate. One possible choice for SΓ is in this case the field of rational functions C (z). But one could equally well take the field of meromorphic functions
of any domain of the complex plane which contains S1 . We see that there is a
continuum of possible choices.
But in fact there are two natural candidates for the ring SΓ: the rational closure
R(CΓ ⊂ UΓ) of CΓ in UΓ and the division closure D(CΓ ⊂ UΓ) of CΓ in UΓ.
They constitute a kind of minimal possible choice for SΓ. We will discuss these
rings in detail in Section 8 and in Appendix III.
Finally we should mention the following result which is implicit in [48].
Theorem 5.16. Suppose Γ is torsionfree, then the Atiyah conjecture with coefficients in R holds for Γ if and only if the division closure D(RΓ ⊂ UΓ) of RΓ
in UΓ is a skew field.
Proof. This will be proven as Proposition 8.30 in Section 8.
5.3
The Relation to the Isomorphism Conjecture in Algebraic K-theory
The map iΓ in condition (B) of Theorem 5.10 factorizes as follows:
M
K0 (CH)
H∈F inΓ
iΓ
K0 (SΓ)
6
?
jΓ
colimH∈F inΓ K0 (CH) - K0 (CΓ)
32
The map on the left is (by construction) always surjective. In this subsection we
will explain why a variant of the isomorphism conjecture in algebraic K-theory
([27], [19]) implies that the bottom map is an isomorphism.
To form the colimit we understand F inΓ as a category with morphisms given
by inclusions K ⊂ H and conjugation maps cg : H → g H, h 7→ ghg −1 with
g ∈ Γ. These maps induce ring homomorphisms of the corresponding group
algebras and K0 (C ? ) can be considered as a functor from F inΓ to the category
of abelian groups.
Equivalently we could work with the orbit category: The orbit category OrΓ
has as objects the homogeneous spaces Γ/H, and morphisms are Γ-equivariant
maps. Let Or(Γ, F in) be the full subcategory whose objects are the homogeneous spaces Γ/H with H finite. There is an obvious bijection between the set
of objects of F inΓ and Or(Γ, F in) but the morphisms differ. This difference
vanishes on the level of K-theory since an inner automorphism ch : H → H
with h ∈ H induces the identity map on K0 .
colimF inΓ K0 (C ? ) = colimOr(Γ,F in) K0 (C ? ).
One possible formulation of the isomorphism conjecture is as follows:
Conjecture 5.17. The assembly map
KCOrΓ
(E(Γ, F in))
∗
- K∗ (CΓ)
Ass
is an isomorphism.
We will not explain the construction of the assembly map. The interested reader
should consult [19]. But let us say a few words about the source and target of
the assembly map to get an idea what it is about.
The target: The target of the assembly map consists of the algebraic K-groups
Kn (CΓ), n ∈ Z of the group ring CΓ. The isomorphism conjecture should be
seen as a means to compute these groups.
The source: KCOrΓ
( − ) is a generalized equivariant homology theory similar
∗
to Bredon homology (see [6]), but with more complicated coefficients. E(Γ, F in)
is the classifying space for the family of finite subgroups [23, I.6], i.e. it is a ΓCW complex which is uniquely determined up to Γ-equivariant homotopy by
the following property of its fixed point sets.
∅
H∈
/ FinΓ
.
E(Γ, F in)H =
contractible H ∈ FinΓ
There is an equivariant version of an Atiyah-Hirzebruch spectral sequence [19,
Section 4] which helps to actually compute the source of the assembly map.
Hp (C∗ (E(Γ, F in)? )) ⊗ZOrΓ Kq (C ? )) ⇒ KCOrΓ
∗ (E(Γ, F in)).
Here C∗ (E(Γ, F in)? ) is the cellular chain complex of the space E(Γ, F in) viewed
as a functor from the category OrΓ to the category of chain complexes by sending
33
Γ/H to the cellular chain complex of the fixed point set. The tensor product is
a tensor product over the orbit category [50, 9.12 on page 166]. In analogy to
ordinary group homology one should think of the ZOrΓ-chain complex as a free
resolution of the ZOrΓ-module ZF in ( ? ) which is given by
0
H∈
/ FinΓ
ZF in (H) =
.
Z
H ∈ FinΓ
We see that since E(Γ, F in)H = ∅ for infinite H only the K∗ (CH) for H finite
enter the computation. So the isomorphism conjecture is a device to compute
K∗ (CΓ) from the knowledge of the K-theory of the finite subgroups. How this
information is assembled is encoded in the space E(Γ, F in).
Note 5.18. Suppose the isomorphism conjecture 5.17 is true. Then
(i) The negative K-groups K−i (CΓ) vanish.
∼
=
(ii) There is an isomorphism colimK∈F inΓ K0 (CK) → K0 (CΓ).
Proof. For a finite group H the group algebra CH is semisimple, and in particular it is a regular ring. (Here regular does not mean von Neumann regular!).
It is known that the negative K-groups for such rings vanish [74, page 154]. We
see that the above spectral sequence becomes a first quadrant spectral sequence
and in particular K−i (CΓ) = 0. Moreover at the origin (p, q) = (0, 0) there are
no differentials and no extension problems to solve. We see that
K0 (CΓ) ∼
= H0 (C∗ (E(Γ, F in)? ) ⊗ZOrΓ K0 (C ? )).
Since the ZOrΓ-chain complex C∗ (E(Γ, F in)? ) is a free resolution of ZF in ( ? )
and the tensor product is right exact [50, 9.23 on page 169] we get
H0 (C∗ (E(Γ, F in)? ) ⊗ZOrΓ K0 (C ? )) ∼
= ZF inΓ ( ? ) ⊗ZOrΓ K0 (C ? ).
From the definition of the tensor product over the orbit category we get
ZF inΓ ( ? ) ⊗ZOrΓ K0 (C ? ) = colimF inΓ K0 (C ? ).
The map which gives the isomorphism in 5.18(ii) should coincide with the natural map jΓ above.
Proposition 5.19. The map
jΓ : colimK∈F inΓ K0 (CK) → K0 (CΓ)
is always rationally injective, i.e. jΓ ⊗ idQ is injective.
34
Proof. The proof uses the Hattori-Stallings rank which is also known as the Bass
trace map. For a ring R denote by [R, R] the subgroup of the additive group of R
generated by commutators, i.e. elements of the form ab − ba. If R is a C-algebra
this subgroup is a complex vector space. Every additive map f out of R which
has the trace property f (ab − ba) = 0 factorizes over HC0 (R) = R/ [R, R]. The
natural map R → HC0 (R), a 7→ a extends to matrices via
X
(aij ) 7→
aii .
i
This extension still has the trace property and is compatible with stabilization.
It leads therefore to a map
ch0 : K0 (R) → HC0 (R).
This is the Hattori-Stallings rank, which can also be considered as a Chern character [49, Section 8.3]. HC0 is functorial and ch0 is a natural transformation.
We obtain a commutative diagram
colimH∈F inΓ K0 (CH)
jΓ-
K0 (CΓ)
f
?
?
colimH∈F inΓ HC0 (CH) - HC0 (CΓ)
Note that it is sufficient to show that jΓ ⊗ C is injective. We claim that the
bottom map in the above diagram is injective and the complex linear extension
fC of f is an isomorphism. Given a group G we denote by conG the set of
conjugacy classes of G. The natural quotient map G → conG induces a map
of free vector spaces CG → CconG. One checks that the kernel of this map is
[CG, CG] and therefore HC0 (CG) can be naturally (!) identified with CconG.
Since the functor free vector space is left adjoint to the forgetful functor it
commutes with colimits. Therefore, in order to show that the bottom map is
injective it is sufficient to show that the map
colimK∈F inΓ conK → conΓ
is injective. Given two elements h and k with h = gkg −1 for some g ∈ Γ the conjugation map cg is a map in the category F inΓ and therefore the corresponding
conjugacy classes are already identified in the colimit.
To show that fC is an isomorphism it is sufficient to show that for every finite
group H the map
ch0 (H)C : K0 (CH) ⊗Z C → HC0 (CH)
is an isomorphism because
colimH∈F inΓ (K0 (CH) ⊗Z C) ∼
= colimH∈F inΓ (K0 (CH)) ⊗Z C.
35
A finitely generated projective CH-module is a finite dimensional complex Hrepresentation. K0 (CH) can be identified with the complex representation ring
and mapping a representation V to its character χV gives a map
K0 (CH) → CconH.
Now the projection corresponding to an isotypical summand W of the left regular representation is given by
pW =
dimC W X
χW (h)h ∈ CH,
|H|
h∈H
compare [79]. The χW are integer multiples of a basis of the free abelian group
K0 (CH). The above formula tells us that up to a base change g 7→ g −1 and
constants, the class [W ] is mapped to χW under the Hattori-Stallings rank
ch0 (H). It is well known that the χW constitute a basis of CconH. We see that
ch0 (H)C is an isomorphism.
Note that the map f in the proof above is in general not injective. In fact
there are examples of groups where jΓ is known to be surjective, but K0 (CΓ)
contains torsion [44]. In these cases injectivity of f would imply that jΓ is an
isomorphism, but f maps to a vector space and can therefore not be injective.
The original isomorphism conjecture [27] deals with the integral group ring ZΓ.
In that case the family of finite subgroups has to be replaced by the larger
family of virtually cyclic subgroups. It seems that much more is known about
the isomorphism conjecture for ZΓ because the K-groups of integral group rings
are more directly related to topology.
In our case the following gives some evidence for the isomorphism conjecture.
The first statement is a special case of Moody’s induction theorem.
Theorem 5.20.
map
(i) Let Γ be a polycyclic-by-finite group, then the natural
jΓ : colimH∈F inΓ K0 (CH) → K0 (CΓ)
is surjective and rationally injective.
(ii) The same holds for virtually free groups.
Proof. (i) For a virtually polycyclic group Γ the group ring is noetherian (see
8.12) and of finite global dimension [77, Theorem 8.2.20]. Therefore the category
of finitely generated modules coincides with the category of modules which admit
a resolution of finite length by finitely generated projective modules. By the
resolution theorem [74, 3.1.13] we get that the natural map K0 (CΓ) → G0 (CΓ)
is an isomorphism. Now the above follows from the general formulation of
Moody’s induction theorem, see Theorem 8.22, and the above Proposition 5.19.
(ii) See [48, Lemma 4.8.]. Note that a virtually free group is always a finite
extension of a free group.
36
6
Atiyah’s Conjecture for the Free Group on
Two Generators
In this section we will prove the Atiyah conjecture in the case where Γ = Z ∗ Z is
a free group on two generators. The proof uses the notion of Fredholm modules
which in this form goes back to Mishchenko [59]. The first example in Connes’s
programme for a non-commutative geometry in [17] is a new conceptional proof
of the Kadison conjecture for the free group using Fredholm modules. The
geometry of the particular Fredholm module used in that case was clarified by
Julg and Valette in [39]. The Kadison conjecture says that the reduced C ∗ ∗
algebra Cred
(Γ) of a torsionfree group has no nontrivial projections. In [48]
Linnell observed that the same technique applies to give a proof of Atiyah’s
conjecture. We will start with some notions from non-commutative geometry.
6.1
Fredholm Modules
Let H be a Hilbert space and B(H) be the algebra of bounded linear operators
on that Hilbert space. We denote by L0 (H) and L1 (H) the ideal of operators of
finite rank respectively the ideal of trace class operators. For p ∈ [1, ∞) we have
the Schatten ideals Lp (H). The usual trace of a trace class operator a ∈ L1 (H)
is denoted by tr(a). The ideal of compact operators is denoted by K. Let B
be a ∗-algebra. A B-representation on H is by definition a ∗-homomorphism
ρ : B → B(H). We explicitly allow that ρ(1) 6= idH .
For our purposes we use a slightly modified notion of Fredholm modules. To
stress the difference let us start with a definition which is ideal for theoretical
purposes, but unfortunately not very realistic.
Definition 6.1. A (1-summable) Fredholm module (H, ρ+ , ρ− ) consists of two
representations ρ± : B → B(H), such that ρ+ (b) − ρ− (b) ∈ L1 (H) for all b ∈ B.
For us a Fredholm module for B is basically a tool designed to produce a homomorphism τ : K0 (B) → R. Namely one verifies (see 6.3 below) that
X
τ : K0 (B) → R p = (pij ) 7→
tr(ρ+ (pii ) − ρ− (pii ))
i
defines such a homomorphism. It may be hard or even impossible to construct
∗
a nontrivial Fredholm module in this sense for B = Cred
Γ or B = N Γ, compare
[18]. To attack the Kadison conjecture one is more modest and restricts oneself
∗
Γ which is closed under holomorphic function
to a dense subalgebra B ⊂ Cred
∗
Γ).
calculus, because for such an algebra it is known that K0 (B) = K0 (Cred
For purposes of the Atiyah conjecture we do not need B to be in any sense dense
in N Γ, but we have to make sure that we can apply τ to all the projections we are
interested in, namely those of the form pker a for a ∈ ZΓ. The following definition
is therefore suitable for our purposes. We will see later that 0-summability is
crucial in order to show that we can handle the relevant projections.
37
Definition 6.2. Let A be a von Neumann algebra and B ⊂ A an arbitrary ∗closed subalgebra. A p-summable (A, B)-Fredholm module (H, ρ+ , ρ− ) consists
of two A-representations ρ± : A → B(H), such that ρ+ (a) − ρ− (a) ∈ Lp (H) for
all a ∈ B.
In the next subsection we will construct a 0-summable (N Γ, CΓ)-Fredholm module. Note that the task of finding a p-summable module becomes more difficult
when p becomes smaller. But once we have such a module the following holds.
Proposition 6.3. Let ρ± : A → B(H) be a p-summable (A, B)-Fredholm module.
(i) For q ∈ {0} ∪ [1; ∞) the set Aq = {a ∈ A | ρ+ (a) − ρ− (a) ∈ Lq (H)} is a
0
∗-subalgebra of A and we have inclusions A0 ⊂ A1 ⊂ Aq ⊂ Aq ⊂ A for
q ≤ q0 .
(ii) If the Fredholm module is p-summable, then B ⊂ Ap .
(iii) The linear map
τ : A1 → C
a 7→ tr(ρ+ (a) − ρ− (a))
has the trace property τ (ab) = τ (ba) for all a, b ∈ A1 .
(iv) Tensoring the (A, B)-Fredholm module ρ± with the standard representation
ρ0 of Mn (C) on Cn yields an (Mn (A), Mn (B))-Fredholm module. We have
Mn (Ap ) ⊂ Mn (A)p and ρ± ⊗ ρ0 is again p-summable.
(v) The map τ extends to matrix algebras by
τ : Mn (A1 ) → C
(aij ) 7→
X
τ (aii )
i
and yields a homomorphism
τ : K0 (A1 ) → R,
p 7→ τ (p).
Proof. Since Lp (H) is an ideal the first statement follows from
ρ(ab) − ρ− (ab) = ρ+ (a) [ρ+ (b) − ρ− (b)] + [ρ+ (a) − ρ− (a)] ρ− (b).
If we now apply tr to this equation and use that tr(cd) = tr(dc) for a trace class
operator c ∈ L1 (H) we see that τ indeed has the trace property. The statement
about matrix algebras follows from Lp (H) ⊗ Mn (C) ⊂ Lp (H ⊗ Cn ) and the
definitions.
Since we want to show that certain numbers are integers the following observation about the difference of projections will become a key ingredient.
Lemma 6.4. Let p and q ∈ B(H) be two projections for which p − q ∈ L1 (H),
then tr(p − q) is an integer.
38
Proof. The key observation is that there is a selfadjoint compact operator which
commutes with p and q at the same time, namely (p − q)2 . Let
M
Eλ ⊕ ker(p − q)2
H=
λ6=0
be a decomposition of H into the Hilbert sum of eigenspaces of (p − q)2 . Since p
and q commute with (p− q)2 they both respect this decomposition. We will now
compute the trace of (p − q) with respect to a basis which is compatible with
the above decomposition. Since ker(p − q) = ker(p − q)2 only the eigenspaces
Eλ for eigenvalues λ 6= 0 contribute. Since they are all finite dimensional we get
X
X
tr(p − q) =
tr((p − q)|Eλ ) =
tr(p|Eλ ) − tr(q|Eλ ).
λ6=0
λ6=0
On the right hand side we have differences of dimensions of vector spaces and
therefore integers. Since the left hand side is finite, only finitely many of these
differences can be nonzero.
In our situation we get:
Corollary 6.5. If (H, ρ+ , ρ− ) is a Fredholm module and if p ∈ A1 is a projection, then τ (p) is an integer. The image of τ : K0 (A1 ) → R lies in Z.
Proof. ρ+ (p) and ρ− (p) are projections, and by definition of A1 the difference
ρ+ (p) − ρ− (p) is a trace class operator. The above lemma applies.
For an operator a ∈ B(H) denote the orthogonal projection onto its kernel by
pker a . The passage from a to pker a is of particular importance for the Atiyah
conjecture. The following lemma explains why we are so much interested in
0-summable Fredholm modules.
Lemma 6.6. Let a, b ∈ B(H) be operators with a − b ∈ L0 (H), then also
pker a − pker b ∈ L0 (H).
Proof. Let K = ker(a − b) and let K ⊥ be the orthogonal complement in H. We
have a|K = b|K and ker a ∩ K = ker b ∩ K and therefore also pker a |ker a∩K =
pker b |ker a∩K . Now let K 0 be an orthogonal complement of ker a ∩ K in K,
then K 0 is perpendicular to ker a and ker b, and thus we have pker a |K 0 = 0 =
pker b |K 0 . We see that pker a and pker b differ only on the space K ⊥ which is finite
dimensional by assumption. In particular pker a − pker b ∈ L0 (H).
6.2
Some Geometric Properties of the Free Group
Let Γ = Z∗ Z =< s, t | > be a free group on two generators s and t. The Cayley
graph with respect to this set of generators is a tree. The group Γ operates on
this tree. On the set of vertices V the operation is free and transitive, on the
set E of edges the operation is also free but there are as many orbits as there
39
are generators. Therefore there are isomorphisms V ∼
= Γ and E ∼
= Γ t Γ of Γsets. The tree is turned into a metric space by identifying each edge with a unit
interval. As a metric space this tree has the pleasant property, that given any
two distinct points x 6= y there exists a unique geodesic joining the two. We will
denote this geodesic symbolically by x → y. Now given a geodesic x → y there
is a unique edge Init(x → y) ∈ E, the initial edge, which meets the geodesic
and x at the same time. This allows us to define the following map
f :V
x
x0
→ E t {pt}
7→ Init(x → x0 ) if x 6= x0
7→ pt.
Here x0 ∈ V is an arbitrary but fixed vertex. This map is a bijection and it is
almost Γ-equivariant in the following sense:
Lemma 6.7. For a fixed g ∈ Γ there is only a finite number of vertices x 6= x0
with gf (x) 6= f (gx). The number of exceptions equals the distance from x to
gx0 .
Proof. Since we are dealing with a tree we have gInit(x → x0 ) = Init(gx →
gx0 ) 6= Init(gx → x0 ) if and only if gx ∈ (gx0 → x0 ).
It is shown in [22] that free groups are the only groups which admit maps with
properties similar to those of the map f above.
6.3
Construction of a Fredholm Module
We will now use the map f : V → E t {pt} from the preceding subsection
to construct a 0-summable (N Γ, CΓ)-Fredholm module for which the trace τ
coincides with the standard trace on N Γ we are interested in.
Let l2 V and l2 E be the free Hilbert spaces on the sets V and E. From the
preceding subsection we know that we have isomorphisms of CΓ-representations
l2V ∼
= l2 Γ and l2 E ∼
= l2 Γ ⊕ l2 Γ. Therefore both representations extend to N Γrepresentations. Let C denote the very trivial one-dimensional representation
with ax = 0 for all a ∈ N Γ and x ∈ C. If we consider the map f above as a
bijection of orthonormal bases (where {pt} is the basis for C) we get an isometric
isomorphism of Hilbert spaces
F : l2 V → l2 E ⊕ C.
Let ρ+ : N Γ → B(l2 V ) be the representation on l2 V and ρ be the one on l2 E⊕C.
We define ρ+ as the pull back of ρ to l2 V via F , i.e. ρ− (a) = F −1 ρ(a)F .
Proposition 6.8. (i) The representations ρ+ and ρ− define a 0-summable
(N Γ, CΓ)-Fredholm module.
(ii) For all a ∈ N Γ we have
trN Γ (a) =
X
x∈V
< (ρ+ (a) − ρ− (a))x, x > .
40
(iii) For a ∈ Mn (N Γ1 ) we have trN Γ (a) = τ (a).
Proof. We start with the second statement. Since Γ operates freely on V and
E we get for g ∈ Γ
< [ρ+ (g) − ρ− (g)] x0 , x0 >
= < ρ+ (g)x0 , x0 > − < ρ(g)f (x0 ), f (x0 ) >
= δg,e = trN Γ (g)
and for x ∈ V with x 6= x0
< [ρ+ (g) − ρ− (g)] x, x >
= < ρ+ (g)x, x > − < ρ(g)f (x), f (x) >
= δg,e − δg,e = 0.
Summing over x ∈ E we get the claim first for g ∈ Γ and then by linearity for a ∈
CΓ. The claim follows if we can show that trN Γ (a) and < [ρ+ (a) − ρ− (a)] x, x >
are continuous functionals with respect to the weak (operator) topology on N Γ,
because N Γ is the weak closure of CΓ. Now trN Γ (a) =< ae, e > is weakly
continuous, since | < . e, e > | is one of the defining seminorms for the weak
topology. One easily verifies that ρ+ and ρ and therefore also ρ− and ρ+ − ρ−
are weak-weak continuous. Since | < . x, x > | is again a defining seminorm for
the weak topology the claim follows.
For the first statement we have to show that for every a ∈ CΓ the operator
ρ+ (a) − ρ− (a) has finite rank. By linearity it is sufficient to treat ρ+ (g) − ρ− (g)
for g ∈ Γ. By Lemma 6.7 we have
ρ+ (g)x = gx = f −1 (gf (x)) = (F −1 ◦ ρ(g) ◦ F )x = ρ− (g)x
for x ∈ V with finitely many exceptions.
The last statement follows immediately from the second.
We need one further lemma.
Lemma 6.9. If a lies in N Γ0 , then also the projection pker a lies in N Γ0 .
Proof. By definition of N Γ0 we know that ρ+ (a)−ρ− (a) is a finite rank operator.
From 6.6 we know that also pker ρ+ (a) − pker ρ− (b) is of finite rank. Since up to
the degenerate summand C we are dealing with direct sums of the left regular
representation we get
ρ+ (pker a ) = pker ρ+ (a)
respectively ρ− (pker a ) + pC = pker ρ− (a)
where pC is the projection onto C. The claim follows.
Theorem 6.10. Let Γ = Z ∗ Z be the free group on two generators and a ∈
Mn (CΓ). Let pker a denote the orthogonal projection onto ker a ⊂ l2 Γn , then
trΓ (pker a ) is an integer.
41
Proof. By 0-summability we know CΓ ⊂ N Γ0 and therefore
Mn (CΓ) ⊂ Mn (N Γ0 ) ⊂ Mn (N Γ)0 .
Let a be in Mn (CΓ). From the matrix analogue of the preceding lemma we know
that pker a ∈ Mn (N Γ)0 . By Proposition 6.8 we know that trN Γ (pker a ) = τ (pker a )
which is an integer by Corollary 6.5.
Corollary 6.11. The Atiyah conjecture with complex coefficients holds for the
free group on two generators.
Proof. We have dimU (ker(a ⊗ idU Γ )) = trA (pker a ) by definition of dimU .
42
7
Classes of Groups and Induction Principles
The Atiyah conjecture is proven in [48] for groups in a certain class C (Definition
7.5), which in addition have a bound on the orders of finite subgroups. In this
section we will review the class C and prove that it is closed under certain
processes (Theorem 7.7). The class C contains the class of elementary amenable
groups. So we begin by collecting some facts about elementary amenable groups
and the related class of amenable groups.
7.1
Elementary Amenable Groups
Definition 7.1. The class EG of elementary amenable groups is the smallest
class of groups with the following properties:
(i) EG contains all finite groups and all abelian groups.
S
(ii) EG is closed under directed unions: If Γ = i∈I Γi , where the Γi , i ∈ I
form a directed system of subgroups (given i, j ∈ I there exists k ∈ I, such
that Γi ⊂ Γk and Γj ⊂ Γk ) and each Γi belongs to EG, then Γ belongs to
EG.
(iii) EG is closed under extensions: If 1 → H → Γ → G → 1 is a short exact
sequence such that H and G are in EG, then Γ is in EG.
(iv) EG is closed under taking subgroups.
(v) EG is closed under taking factor groups.
It is shown by Chou in [12] that the first three properties suffice to characterize
the class EG. That EG is closed under taking subgroups and factor groups
follows from the other properties. Chou also gives a description of the class of
elementary amenable groups via transfinite induction. Following [43] we will
give a similar description below. But first we need some notation.
If X and Y are classes of groups, let X Y denote the class of X -by-Y groups,
that means the class of those groups Γ, for which there is an exact sequence
1 → H → Γ → G → 1 with H in X and G in Y. Similarly let LX denote
the class of locally-X -groups: A group Γ is in LX if every finitely generated
subgroup of Γ is in X .
Proposition 7.2. The class of elementary amenable groups has the following
description via transfinite induction: Let B denote the class of finitely generated abelian-by-finite groups and let for every ordinal α the class Dα be defined
inductively as follows:
• D0 = 1.
• Dα = (LDα−1 )B if α is a successor ordinal.
S
• Dα = β<α Dβ if α is a limit ordinal.
43
S
The class of elementary amenable groups EG coincides with α≥0 Dα . Moreover
every class Dα is subgroup closed and closed under extensions by finite groups.
Proof. See Lemma 3.1 in [43].
Whenever one wants to prove a statement about elementary amenable groups
one can use such a description to give an inductive proof. It is therefore desirable
to have small induction steps. On the other hand it might be useful if the
intermediate classes Dα have good properties, as for example being subgroupclosed or being closed under extensions by finite groups. In easy cases such an
inductive proof comes down to the following.
Proposition 7.3 (Induction Principle for EG). Suppose P is a property of
groups such that:
(I) P holds for the trivial group.
(II) If 1 → H → Γ → A → 1 is an exact sequence, where A is finitely generated
abelian-by-finite and P holds for H, then it also holds for Γ.
S
(III) If Γ = i∈I Γi is the directed union of the subgroups Γi and P holds for
all the Γi , then P holds for Γ.
Then P holds for all elementary amenable groups. Instead of (II) it would also
be sufficient to verify:
(II)’ If 1 → H → Γ → A → 1 is an exact sequence, where A is finite or cyclic,
and P holds for H, then it also holds for Γ.
Proof. This follows easily via transfinite induction from the description of the
class given in Proposition 7.2. Since one extension by a finitely generated
abelian-by-finite group can always be replaced by finitely many extensions by
infinite cyclic groups followed by one extension by a finite group (II)’ implies
(II).
Before we go on and introduce Linnell’s class C, we will discuss the class of
amenable groups.
Definition 7.4. A group Γ is called amenable, if there is a Γ-invariant linear
map µ : l∞ (Γ, R) → R, such that
inf{f (γ)|γ ∈ Γ} ≤ µ(f ) ≤ sup{f (γ)|γ ∈ Γ} for
f ∈ l∞ (Γ).
Here l∞ (Γ, R) is the space of bounded real valued functions on Γ. The map µ
is called an invariant mean. The class of amenable groups is denoted by AG.
The class of amenable groups AG has the five properties that were used to
define the class of elementary amenable groups [20], therefore EG ⊂ AG. The
simplest example of a group which is not amenable is the free group on two
generators [71, p.6]. Let N F denote the class of groups which do not contain a
free subgroup on two generators. Since AG is subgroup closed AG ⊂ N F . Day
44
posed the question whether EG = AG or AG = N F . The question remained
open for a long time. Meanwhile there are examples of groups (even finitely
presented) showing EG =
6 AG due to Grigorchuk [32]. Examples by Olshanski
[68] show AG 6= N F . These examples are finitely generated. Whether there
is a finitely presented group which is not amenable, but does not contain a free
subgroup on two generators, is an open question.
There are many other characterizations of amenable groups. In particular it
seems that amenability of a group Γ has a strong impact on the different algebras
of operators associated to the group. Here are some facts. For more details and
references see [71].
(i) The invariant mean in the definition of amenability can as well be characterized as a Γ-invariant bounded linear functional µ on l∞ (Γ, C) equipped
with supremum-norm, such that µ(1) = 1 and µ(f ) ≥ 0 for every f ≥ 0.
In the language of operator algebras such a µ is called a (Γ-invariant) state
on the commutative C ∗ -algebra l∞ (Γ, C).
(ii) There is the so called Fölner criterion which characterizes amenability
of a group in terms of the growth of its Cayley graph, see [28] and [3,
Theorem F.6.8, p.308].
∗
∗
(iii) The natural map Cmax
(Γ) → Cred
(Γ) from the maximal C ∗ -algebra to
∗
the reduced C -algebra of the group is an isomorphism if and only if the
group Γ is amenable, see [72, Theorem 7.3.9, p. 243].
(iv) There is a notion of amenability for von Neumann algebras and C ∗ algebras. Group von Neumann algebras and group C ∗ -algebras of amenable
groups are special examples ([71, 1.31 and 2.35]).
(v) If Γ is amenable the group von Neumann algebra N Γ is very close to being
flat over the group ring CΓ, namely it is shown in [52, Theorem 5.1] that
dimN Γ TorpCΓ (N Γ, M ) = 0 for all p ≥ 1. Compare also Theorem 8.4.
7.2
Linnell’s Class C
In [48] Linnell introduced the following class of groups:
Definition 7.5. Let C denote the smallest class of groups, with the following
properties:
(i) C contains all free groups.
(ii) C is closed under extensions by elementary amenable groups: If there is a
short exact sequence 1 → H → Γ → G → 1 with H in C and G in EG,
then Γ belongs to C.
S
(iii) C is closed under directed unions: If Γ = i∈I Γi , where the Γi , i ∈ I form
a directed system of subgroups (given i, j ∈ I there exists k ∈ I, such that
Γi ⊂ Γk and Γj ⊂ Γk ) and each Γi belongs to C, then Γ belongs to C.
45
The class of groups in C which have a bound on the orders of finite subgroups
will be denoted by C 0 .
Again there is an inductive description of this class of groups. Concerning
notation compare the remarks following Definition 7.1.
Proposition 7.6. The class C has the following description via transfinite induction: Let B denote the class of finitely generated abelian-by-finite groups
and let for every ordinal α the class Cα be defined inductively as follows:
• C0 is the class of free-by-finite groups.
• Cα = (LCα−1 )B if α is a successor ordinal.
S
• Cα = β<α Cβ if α is a limit ordinal.
S
The class C coincides with α≥0 Cα . Moreover, every class Cα is subgroup closed
and closed under extensions by finite groups.
Proof. See [48].
Given a concrete group it is often difficult to tell whether or not it belongs to
the class C. It is therefore useful to know that C is closed under other processes
than the defining ones.
Theorem 7.7.
(i) The class C is closed under taking subgroups.
(ii) If Γ is a group in C and A is an elementary amenable normal subgroup,
then the quotient Γ/A is in C.
(iii) If 1 → Z → Γ → G → 1 is an exact sequence of groups where G is in
C and Z is finite or cyclic, then Γ is in C. More generally, this holds if
Z is any elementary amenable group whose automorphism group is also
elementary amenable.
(iv) The class C is closed under taking arbitrary free products: If Γi , i ∈ I is
any collection of groups in C, then the free product
Γ := ∗i∈I Γi
also lies in C. If all the factors have a bound on the orders of finite
subgroups (Γi ∈ C 0 ) and if there are only finitely many factors, then Γ is
in C 0 .
This allows us to give some concrete examples of interesting groups in the class
C.
Example 7.8. It is well known that P SL2 (Z) ∼
= Z/2Z ∗ Z/3Z. Theorem 7.7
immediately yields that P SL2 (Z) lies in the class C and also that SL2 (Z) and
GL2 (Z) are in C.
46
The rest of this section will be concerned with the proof of Theorem 7.7. Part
(i) follows immediately from the fact that all the Cα in Proposition 7.6 are closed
under taking subgroups. The proof of (ii), (iii) and (iv) is based on the following
induction principle which will be used repeatedly.
Proposition 7.9 (Induction Principle for C). Suppose P is a property of
groups such that:
(I) P holds for free groups.
(II) If there is a short exact sequence 1 → H → Γ → A → 1 with A elementary
amenable and if the property holds for H, then it is also holds for Γ.
(III) If Γ is the directed union of subgroups Γi and the property holds for every
Γi , then it holds for Γ.
In this case P holds for every group in the class C. Instead of (II) it is also
sufficient to check one of the following statements:
(II)’ If there is a short exact sequence 1 → H → Γ → A → 1 with A finitely
generated abelian-by-finite and if the property holds for H, then it also
holds for Γ.
(II)” If there is a short exact sequence 1 → H → Γ → A → 1 with A finite or
infinite cyclic and if the property holds for H, then then it also holds for
Γ.
Proof. This follows in the (II)’-case by transfinite induction from the description of the class C given in Proposition 7.6. Since (II) is stronger than (II)’
it remains to be shown that also (II)” is sufficient. This follows from the fact
that one can split an extension by one finitely generated abelian-by-finite group
into finitely many extensions by infinite cyclic groups followed by one extension
by a finite group.
Proof of 7.7(ii). We use the induction principle. Let P (Γ) be the following
statement: For every elementary amenable normal subgroup A / Γ the quotient
Γ/A is in C.
(I) Let Γ be a free group. Since any subgroup of a free group is free, an elementary amenable subgroup A must be either trivial or infinite cyclic. Suppose A
is infinite cyclic. Since every finitely generated normal subgroup of a free group
has finite index ([57, Chapter I, Proposition 3.12.]), the group Γ/A is finite and
therefore in C.
(II) Let 1 → H → Γ → A0 → 1 be an exact sequence with A0 elementary
amenable and suppose P (H) holds. Let A be an elementary amenable normal
subgroup of Γ. The following diagram is commutative and has exact rows and
47
columns.
A∩H
∩
?
A⊂
⊂
- H
- H/A ∩ H
∩
∩
?
- Γ
?
- Γ/A
?
?
A0
?
?
A0
=
As a subgroup of A, the group A ∩ H is elementary amenable. By P (H) the
group H/A ∩ H is in C. Since Γ/A is an extension of H/A ∩ H by the elementary
amenable groupSA0 we see, that Γ/A is in C.
(III) Let Γ = i∈I Γi be a directed union, suppose P (Γi ) holds for all i ∈ I
and let A / Γ be an elementary amenable normal subgroup. The quotient Γ/A
is the directed union of the Γi /A ∩ Γi which are in C since A ∩ Γi is elementary
amenable as a subgroup of A and P (Γi ) holds by assumption. This completes
the proof of 7.7 (ii).
Proof of Theorem 7.7(iii). The proof of is very similar. Let now P (G) be the
p
- G → 1 is an exact sequence with Z and if
statement: If 1 → Z → Γ
Aut(Z) is elementary amenable, then Γ is in C.
(I) If G is a free group, then the sequence splits. A section determines a
homomorphism α : G → Aut(Z) and Γ is a semidirect product Γ ∼
= Z oα G. In
the commutative diagram
im(α)
α◦p
Z
⊂
Z
⊂
- Γ
6
∪
- ker(α ◦ p)
6
6
α
p - G
6
∪
- ker(α)
the group ker(α ◦ p) is a direct product of Z and the free group ker(α) and is
therefore in C. Now Γ is an extension of ker(α ◦ p) by im(α) ⊂ Aut(Z), which
is by assumption elementary amenable. We conclude that Γ is in C.
(II) Let 1 → H → G → A → 1 be an exact sequence with A elementary
48
amenable and suppose P (H) holds. In the commutative diagram
Z
=
∩
p
−1
?
(H)
Z
∩
⊂
?
?
H⊂
?
- Γ
- A
?
?
- G
- A
the group p−1 (H) is in C because P (H) holds, and Γ is an extension of this
group by an elementary amenable group. Therefore Γ is in C and P (G) holds.
(III) Let G be the directed union of the groups Gi , i ∈ I. In the exact sequence
1 → Z → p−1 (Gi ) → Gi → 1 the middle group is in C by the assumption P (Gi ).
Now Γ is the directed union of the p−1 (Gi ) and is therefore in C. This finishes
the proof of 7.7 (iii).
We will now go on to prove the statement (iv) about free products in Theorem
7.7. We need the following observation about extensions of free products.
p
Lemma 7.10. Let 1 → K → G −
→ Q → 1 be a short exact sequence of groups
and let {gq , q ∈ Q} ⊂ G be a fixed set of representatives for the cosets. Then
for any group H the kernel of the homomorphism
p∗1
G ∗ H −−→ Q
is the free product
(∗q∈Q H gq ) ∗ K.
Here H g is the conjugate of H by g in G∗H. In particular, we have an extension
1 → (∗q∈Q H) ∗ K → G ∗ H → Q → 1
Proof. Note that by definition of a free product every element x ∈ G ∗ H can
be written uniquely as a word x = g1 h1 g2 h2 · . . . · gn hn with letters from G and
H, where g2 , . . . , gn 6= 1 and h1 , . . . , hn−1 6= 1. Every element g ∈ G can be
written uniquely as g = kgq for some k ∈ K and q ∈ Q. Define inductively ki
and qi such that g1 = k1 gq1 , gq1 g2 = k2 gq2 , . . . , gqn−1 gn = kn gqn . This allows
us to rewrite x as follows:
x = k1 (gq1 h1 gq−1
)k2 (gq2 h2 gq−1
)k3 · . . . · kn (gqn hn gq−1
)gqn .
1
2
n
Now suppose x is in the kernel of p ∗ 1. Since p ∗ 1(x) = gqn = 1 we see that x
can be written as a word with letters in the H gq and in K as proposed by the
49
brackets. Suppose there is another such spelling of x as a word with letters in
the H gq and in K
−1 0
−1
0
0
0
x = (k10 gq10 )h01 (gq−1
0 k2 gq 0 )h2 (gq 0 k3 gq 0 ) · . . . · hm (gq 0 ).
2
3
m
1
2
Reading this as indicated by the brackets as a word with letters in G and H and
using the above mentioned uniqueness yields m = n and g1 = k10 gq10 , h01 = h1 ,
−1
0
0
0
g2 = gq−1
= 1. Uniqueness of the spelling as a
0 k2 gq 0 , h2 = h2 , . . . , hn = hn , gq 0
2
n
1
gq
word with letters in the H and K follows inductively from this, making use of
the uniqueness of the ki and qi . This proves that the kernel is the free product
∗q∈Q H gq ∗ K.
Lemma 7.11. Let Γ be in C and F be any free group. Then Γ ∗ F is in C.
Proof. We use the induction principle. (I) The statement is clear if Γ is a free
group. (II) Now take an extension 1 → H → Γ → A → 1 where A is elementary
amenable and H ∗ F 0 belongs to C for any free group F 0 . By Proposition 7.10
we get an extension 1 → ∗a∈A F ∗ H → Γ ∗ F → A → 1. Since F 0 := ∗a∈A F
is again free, this is an extension of a group ofSC by an elementary amenable
group, therefore Γ ∗ F is in C. (III) Let Γ = i∈I Γi be a directed
S union of
subgroups and suppose Γi ∗ F ∈ C for all i ∈ I. Then Γ ∗ F = i∈I Γi ∗ F
belongs to C since C is closed under directed unions.
Lemma 7.12. Suppose G is in C, and let I be any index set. Then GI := ∗i∈I G
is in C.
Proof. Proposition 7.10 with p = idZ shows that there is an extension 1 →
∗z∈Z G → G ∗ Z → Z → 1. By Lemma 7.11 the group G ∗ Z belongs to the class
C. Since C is subgroup closed ∗z∈Z G is in C. Again using that C is subgroup
closed we see that the free product of finitely many copies of G belongs to C.
Since GI = ∗i∈I G is the directed union of free products of finitely many copies
of G we are done.
Lemma 7.13. Fix a group Γ ∈ C. Let I be any index set and GI := ∗i∈I G. If
G is in C, then also Γ ∗ GI .
Proof. Again we use the induction principle. (I) If Γ is a free group, the statement follows from Lemma 7.12 and lemma 7.11. (II) Suppose 1 → H → Γ →
A → 1 is an exact sequence of groups with A elementary amenable. Then we
get an extension 1 → H ∗ GI 0 → Γ ∗ GI → A → 1 with GI 0 := ∗a∈A (∗i∈I G)
another free product of copies of G. If the statement holds for H, then H ∗ GI 0
is in C since we did not specify the index
S set. But then also Γ ∗ GI belongs to
C. (III) If Γ is a directed union Γ = j∈J Γj and Γj ∗ GI is in C for all j ∈ J,
then Γ ∗ GI belongs to C since it is the directed union of the Γj ∗ GI .
Now we are prepared to finish the proof of Theorem 7.7.
50
Proof of Theorem 7.7(iv). Since C is closed under taking subgroups 7.13 implies
that a free product with two factors lies in C if the factors do. Ordinary induction
shows that any finite free product of groups of C belongs to C. Since an arbitrary
product ∗i∈I Γi is the directed union of the subgroups ∗i∈J Γi with J ⊂ I finite,
the group ∗i∈I Γi lies in C.
51
8
Linnell’s Theorem
We will now discuss in detail the following result due to Linnell [48].
Theorem 8.1. If Γ is a group in the class C which has a bound on the orders of
finite subgroups, then the Atiyah conjecture with complex coefficients (see 5.3)
holds for Γ.
For the proof we will follow the strategy outlined in Section 5.2. So our task
is to find intermediate rings of the extension CΓ ⊂ UΓ with good properties.
By definition the class C is closed under taking extensions by finite and infinite
cyclic groups. On the ring theoretical side an extension corresponds to a crossed
product, see Appendix IV. One should think of this as a process which makes the
ring worse, e.g. it raises the homological dimension. The technique to ameliorate
the ring will be localization. In a large part of the following we will therefore
investigate what happens if one combines and iterates these processes. Directed
unions of groups correspond to directed unions of rings and are much easier to
handle.
The candidates for the intermediate rings are the division closure and the rational closure of CΓ in UΓ. The reader who is not familiar with the concepts
of division and rational closure should first read the first three subsections of
Appendix III. In particular the situation of Proposition 13.17 will occur again
and again. We denote by DΓ = D(CΓ ⊂ UΓ) the division closure of CΓ in
UΓ and by RΓ = R(CΓ ⊂ UΓ) the rational closure of CΓ in UΓ. Since a von
Neumann regular ring is division closed and rationally closed (see 13.15) these
rings constitute a minimal possible choice for the ring SΓ we are looking for.
Note 8.2. If the intermediate ring SΓ is von Neumann regular, then we have
DΓ ⊂ RΓ ⊂ SΓ.
We should mention that in all known cases DΓ and RΓ coincide. We denote the
set of elements of CΓ which become invertible in UΓ by T(Γ) = T(CΓ ⊂ UΓ) =
CΓ ∩ UΓ× . Similarly the set of those matrices over CΓ which become invertible
over UΓ is denoted by Σ(Γ) = Σ(CΓ ⊂ UΓ). The set of non-zerodivisors of CΓ
is denoted NZD(CΓ).
Here is now a precise statement of the result that will be proven. Combined
with Theorem 5.10 this implies the above Theorem.
Theorem 8.3. Let Γ be a group in the class C which has a bound on the orders
of finite subgroups, then the following holds.
(A)
The ring DΓ is semisimple, and the inclusion CΓ ⊂ RΓ is universal
Σ(Γ)-inverting.
The semisimplicity of DΓ implies DΓ = RΓ, compare 13.16.
(B)
The natural map
colimK∈F in(Γ) K0 (CK) → K0 (DΓ)
is surjective.
52
With little extra effort we prove the following statement which appears already
in [46].
Theorem 8.4. If moreover Γ is elementary amenable and has a bound on the
orders of finite subgroups, the following holds.
(C)
The set of non-zerodivisors NZD(CΓ) equals T(Γ), i.e. every non-zerodivisor
of CΓ becomes invertible in UΓ. The pair (CΓ, T(Γ)) satisfies the Ore condition. The Ore localization is semisimple and isomorphic to the division
closure:
DΓ ∼
= (CΓ)T(Γ)−1 .
In particular, in this case CΓ ⊂ DΓ is not only universal Σ(Γ)-inverting, but
also universal T(Γ)-inverting. An Ore localization with respect to the set of all
non-zerodivisors is called a classical ring of fractions.
Of course these theorems are proven via transfinite induction. Compare 7.6 and
7.9. Before we give the details of the proof we will discuss the main ingredients
and difficulties. The table below might help the reader to follow the proof. The
statements proven below are labelled according to this table.
Recall that, given classes of groups X and Y, we denote by X Y the class of
X -by-Y groups and by LX the class of locally-X -groups. The subclass X 0 ⊂ X
consists of those groups which have a bound on the orders of finite subgroups.
The class of finite groups is denoted by {finite}. Similarly for other classes of
groups. The statement of (A) above for the group Γ is abbreviated to (A)Γ .
Similarly (A)X , if (A) holds for all groups in the class X . With this shorthand
notation we can summarize the statements proven in the following subsection
as follows:
AIa)
(A){free}
BIb)
(B){free}{finite}
AIIa)
AIIb)
BIIa)+b)
AIII)
BIII)
(A)Y ⇒ (A)Y{infinite cyclic}
(A)Y ⇒ (A)Y{finite}
Y{finite} = Y, (A)Y and (B)Y ⇒ (B)Y{f.g. abelian-by-finite}
(A)Y 0 and (B)Y 0 ⇒ (A)(LY)0
(B)Y ⇒ (B)LY .
We will verify below (page 55) that Theorem 8.3 follows via transfinite induction.
Note that the strong hypothesis in BIIa) + b) forces us to start the induction
with a class of groups which is closed under finite extensions. Therefore it is
not enough to prove (A) and (B) for free groups to start the induction.
53
(A)
I. Starting the induction.
II. Induction step
Y → YB: Extensions by H.
(B)
a) Γ free on two
generators.
b) Γ free by finite.
a)
H infinite
cyclic.
b) H finite.
a) + b) H finitely
generated abelianby-finite.
III. Induction step
Y → LY: Directed
unions.
One might wonder whether one could separate the (A)-part from the (B)-part,
but the proof of AIII) uses the (B)-part of the induction hypothesis. The
(B)-part depends of course heavily on the (A)-part.
That CΓ ⊂ RΓ is universal Σ(Γ)-inverting is only important for the proof
of BIb). If one is only interested in the Atiyah conjecture one could therefore
proceed with the induction steps without this extra statement. We have included
it in the induction since it is interesting in its own right and leads for example to
exact sequences in K-theory, compare Note 10.9. We have tried to give versions
of the statements which use as little hypothesis as possible.
The following remarks summarize where deeper results enter the proof.
AIa) The major part of the work has already been done in Section 6 and relies
on the Fredholm module techniques described there. It is easy to see that
DΓ is a skew field, but to show that CΓ ⊂ DΓ is universal Σ(Γ)-inverting
requires further results about the group ring of the free group.
BIa) This is trivial once we know DΓ is a skew field since K0 (DΓ) = Z.
AIb) Extensions by finite groups cause no problems for the (A)-part.
BIb) Here it is required that CΓ ⊂ DΓ is universal Σ(Γ)-inverting to apply
results of Schofield [78] about the K-theory of universal localizations.
AIIa) This is again easy.
AIIb) This part uses Goldie’s theorem and the structure theory of semisimple
rings. To verify the hypothesis of Goldie’s theorem we have to go back to
operators and functional analysis and use a result of Linnell [47].
AIIa)+b) Combine the two previous parts.
BIIa)+b) Here it does not seem to be possible to prove BIIa) and BIIb) separately.
Moody’s induction theorem is the main ingredient in this part. Moreover
this, part forces us to start the induction with a class of groups that is
closed under extensions by finite groups.
54
AIII) This part is straightforward, but the hypothesis about the bound on the
orders of finite subgroups enters the proof.
BIII) This is easy.
Proof of Theorem 8.3. We use the description of the class C given in Proposition 7.6: Let Γ be in C 0 . Choose a least ordinal α such that Γ ∈ Cα , and suppose
(A) and (B) hold for all Cβ0 with β < α. There are three cases: If α = 0, then
Cα = C0 = {free-by-finite} and the result follows from AIa),
S AIIb) and BIb).
If α is a limit ordinal there is nothing to prove since Cα = β<α Cβ . If α = β + 1
for some ordinal β we have Cα = (LCβ ){f.g. abelian-by-finite} and by assumption (A)C 0 and (B)Cβ0 hold. Now AIII) and BIII) yield (A)(LCβ )0 and (B)(LCβ )0 .
β
Here we used that (LY)0 ⊂ L(Y 0 ) for arbitrary classes Y. Now, since one extension by a finitely generated abelian-by-finite group can always be replaced
by finitely many extensions by infinite cyclic groups followed by one extension
by a finite group we get from AIIa), AIIb) and (Y{f.g. abelian-by-finite})0 =
Y 0 {f.g. abelian-by-finite} that (A)C 0 holds. From Proposition 7.6 we know that
α
Cβ {finite} = Cβ . For an arbitrary class Y{finite} = Y implies (LY){finite} = LY
and we always have Y 0 {finite} = (Y{finite})0 . Therefore BIIa)+b) applies and
yields (B)Cα0 .
We will see that it is relatively easy to prove the following refinements.
CIIa)
(C)Y ⇒ (C)Y{infinite cyclic}
CIIb)
(C)Y ⇒ (C)Y{finite}
CIII)
(C)Y 0 and (B)Y 0 ⇒ (C)(LY)0 .
Theorem 8.4 follows using the induction principle for elementary amenable
groups 7.3. Note that (C) is false for the free group on two generators since
this would imply that DΓ is flat over CΓ and therefore (DΓ is semisimple) UΓ
is flat over CΓ. But this cannot be true since we know that H1 (Γ; UΓ) does not
vanish.
8.1
Induction Step: Extensions - The (A)-Part
In this section we consider an exact sequence of groups
1 → G → Γ → H → 1.
The ring DΓ will be built up in two steps out of DG. We first consider a crossed
product DG ∗ H, and we will show that under certain circumstances DΓ is an
Ore localization of DG ∗ H.
First we collect what we have to know about the behaviour of localizations
with respect to crossed products. For the definition of a crossed product and
our conventions concerning notation we refer to Appendix IV. A ring extension
R ∗ G ⊂ S ∗ G is called compatible with the crossed product structures if the
inclusion map is a crossed product homomorphism. We emphasize again that
Proposition 13.17 is the starting point for most of the considerations below.
55
Proposition 8.5 (Crossed Products and Localization). Let R∗G ⊂ S∗G
be a compatible ring extension of crossed products. The crossed product structure
map on both rings will be denoted by µ : G → R ∗ G ⊂ S ∗ G
(i) The intermediate ring generated by D(R ⊂ S) and µ(G) carries a crossed
product structure such that both inclusions
R ∗ G ⊂ D(R ⊂ S) ∗ G ⊂ S ∗ G
are compatible with the crossed product structures.
(ii) Similarly the ring generated by R(R ⊂ S) and µ(G) in S∗G is a compatible
crossed product.
(iii) Let Σ be a set of matrices over (or elements of ) R which is invariant under
the conjugation maps cµ(g) . Let i : R → RΣ be universal Σ-inverting, then
RΣ ∗ G exists together with a crossed product homomorphism
i ∗ G : R ∗ G → RΣ ∗ G.
This map is universal Σ-inverting.
(iv) If R ⊂ D(R ⊂ S) is universal T(R ⊂ S)-inverting, then
R ∗ G ⊂ D(R ⊂ S) ∗ G
is universal T(R ⊂ S)-inverting.
(v) If R ⊂ R(R ⊂ S) is universal Σ(R ⊂ S)-inverting, then
R ∗ G ⊂ R(R ⊂ S) ∗ G
is universal Σ(R ⊂ S)-inverting.
(vi) Let T ⊂ R be a multiplicatively closed subset which is invariant under the
conjugation maps cµ(g) . Suppose (R, T ) fulfills the right Ore condition,
then (R ∗ G, T ) fulfills the right Ore condition and
RT −1 ∗ G ∼
= (R ∗ G)T −1 .
Proof. (i) Let µ : G → S ∗ G denote the crossed product structure of S ∗ G. By
assumption it differs from the one of R∗G only by the inclusion R∗G ⊂ S∗G. We
have to show that the conjugation maps cµ(g) can be restricted to D = D(R ⊂ S)
and that the ring generated by D and µ(G) is a free D-module with basis µ(G).
For τ there is nothing to be done. By definition D = D(R ⊂ S) is division closed
in S, i.e. D ∩ S × ⊂ D× . Let α : S → S be an automorphism with α(R) = R.
One verifies that α(D) is division closed in S and contains R. Since D is the
smallest ring with these properties we have D ⊂ α(D), and arguing with α−1
we get equality. In particular this applies to cµ(g) . Using this one
P can write
every element in the ring generated by D and µ(G) in the form g∈G dg µ(g)
56
with dg ∈ D. Since S ∗ G is a free S-module with basis µ(G) this representation
is unique.
(ii) The proof is more or less identical with the proof above starting this time
with M(R(R ⊂ S)) ∩ GL(S) ⊂ GL(R(R ⊂ S)).
(iii) Let i : R → RΣ be universal Σ-inverting. By a assumption cµ(g) (Σ) ⊂ Σ
and therefore the composition i ◦ cµ(g) : R → R → RΣ is Σ-inverting. The
universal property gives us a unique extension cµ(g) : RΣ → RΣ of cµ(g) . We
define a multiplication on the free RΣ -module with basis the symbols µ(g) with
g ∈ G by
r µ(g) · r0 µ(g 0 ) = r cµ(g) (r0 ) τ (g, g 0 ) µ(gg 0 )
(1)
for r, r0 ∈ RΣ . Here τ (g, g 0 ) = µ(g) µ(g 0 ) µ(gg 0 )−1 ∈ R× and τ = i ◦ τ . For
associativity we have to check (compare 14.2)
τ (g, g 0 ) · τ (gg 0 , g 00 )
cτ (g,g0 ) ◦ cµ(gg0 )
= cµ(g) (τ (g 0 , g 00 )) · τ (g, g 0 g 00 )
= cµ(g) ◦ cµ(g0 )
(2)
(3)
The first equation follows from associativity in R ∗ G and the fact that cµ(g)
extends cµ(g) . For the second equation note that cτ (g,g0 ) = cµ(g) ◦ cµ(g0 ) ◦ c−1
µ(gg0 ) :
R → R also leaves Σ invariant and has by the universal property an extension
cτ (g,g0 ) : RΣ → RΣ which by uniqueness coincides with cτ (g,g0 ) : RΣ → RΣ .
Arguing with the universal property we verify the second equation. Now as
RΣ ∗ G exists as a ring it makes sense to consider cµ(g) : RΣ ∗ G → RΣ ∗ G. One
checks that cµ(g) restricts to RΣ and cµ(g) = cµ(g) . Similarly the notation τ is
0
0 −1
unambiguous, i.e. (i ◦ τ )(g, g 0 ) = τ (g, g 0 ) = µ(g) µ(g
P) µ(gg ) . P
The map i ∗ G : R ∗ G → RΣ ∗ G given by
rg µ(g) 7→
i(rg )µ(g) is
a Σ-inverting ring homomorphism and compatible with the crossed product
structures. Remains to be verified that it is also universal Σ-inverting. Let
f : R ∗ G → S be a given Σ-inverting homomorphism. Denote by j : R → R ∗ G
and jΣ : RΣ → RΣ ∗ G the natural inclusion. The universal property of
i : R → RΣ applied to the composition f ◦ j : R → R ∗ G → S gives a homomorphism φ0 : RΣ → S with φ0 ◦ i = f ◦ j. Now a ring homomorphism
φ : RΣ ∗ G → S with φ ◦ (i ∗ G) = f and φ ◦ jΣ = φ0 necessarily has to be
defined by φ(jΣ (r)µ(g)) = φ0 (r)f (µ(g)). One ensures that this indeed defines a
ring homomorphism.
(iv) Denote by i : R → RT the universal T inverting homomorphism and by
j : R → D(R ⊂ S) the inclusion. By assumption the homomorphism φ : RT →
D(R ⊂ S) given by the universal property is an isomorphism. By (iii) i ∗ G
exists and is universal T -inverting. From (i) we know that j ∗ G exists and is
T -inverting. The universal property applied to this yields φ ∗ G which is an
isomorphism.
(v) The same argument as in (iii).
(vi) The assumption says that given (ag , s) ∈ (R, T ) there exist (bg , tg ) ∈ (R, T )
−1
such that ag tg = sbg , which becomes s−1 ag = bg t−1
. Moreover finitely
g in RT
many fractions, e.g. (1, cµ(g)−1 (tg )) ∈ (R, T ) can be brought to a common denominator. So there exist (dg , t) ∈ (R, T ) such that cµ(g)−1 (tg )−1 = dg t−1 .
57
Using this one turns
i
hX
s−1
ag µ(g) =
=
X
X
bg t−1
g µ(g)
bg µ(g)cµ(g)−1 (tg )−1 =
hX
i
bg µ(g)dg t−1
into a proof by avoiding denominators. The Ore localization (R ∗ G)T −1 is
universal T -inverting. From (iii) we know that RT ∗ G ∼
= RT −1 ∗ G is also
universal T -inverting. Therefore the two rings are isomorphic.
In particular, we obtain in our situation:
Proposition 8.6 (Extensions and Localization). Let 1 → G → Γ → H →
1 be an exact sequence of groups and let µ be a set theoretical section µ : H → Γ.
(i) The ring DG ∗ H generated by DG and µ(H) in UΓ has a compatible
crossed product structure. The ring generated by RG and µ(H) has a
compatible crossed product structure.
(ii) The crossed product CGT(G) ∗ H exists together with a map
CG ∗ H → CGT(G) ∗ H.
This map is universal T(G)-inverting. Similarly the map
CG ∗ H → CGΣ(G) ∗ H
exists, is compatible and universal Σ(G)-inverting.
(iii) If CG ⊂ DG is universal T(G)-inverting, then
CG ∗ H ⊂ DG ∗ H
is universal T(G)-inverting. If CG ⊂ RG is universal Σ(G)-inverting,
then
CG ∗ H ⊂ RG ∗ H
is universal Σ(G)-inverting.
∼ (CG)T (G)−1 , then
(iv) If (CG, T (G)) fulfills the Ore condition and D(G) =
the pair (CG ∗ H, T(G)) fulfills the Ore condition, CG ∗ H ⊂ DG ∗ H is
universal T(G)-inverting and
DG ∗ H ∼
= (CG ∗ H)T(G)−1
= (CG ∗ H)T(G) ∼
is an Ore localization.
Proof. Apply the previous proposition.
58
The next step is the passage from DG ∗ H to DΓ. We will show that under
certain circumstances this is an Ore localization. Finally we are interested in
combining these two steps. It is therefore important to gather information on
iterated localizations. Note that an analogue of (ii) in the following proposition
for elements instead of matrices does not hold.
Proposition 8.7 (Iterated Localization). Let R ⊂ S be a ring extension.
(i) The division closure and rational closure are in fact closure operations,
namely
and
D(D(R ⊂ S) ⊂ S) = D(R ⊂ S),
R(R(R ⊂ S) ⊂ S) = D(R ⊂ S).
(ii) Let Σ ⊂ M (R) be a set of matrices with Σ ⊂ Σ(R ⊂ S). Let R → RΣ be
universal Σ-inverting and RΣ → (RΣ )Σ(RΣ →S) be universal Σ(RΣ → S)inverting, then the composition of these maps is universal Σ(R ⊂ S)inverting. A little less precise
(RΣ )Σ(RΣ →S) ∼
= RΣ(R⊂S) .
(iii) Let T ⊂ T(R ⊂ S) be a multiplicatively closed subset. Suppose the pair
(R, T ) and the pair (RT −1 , T(RT −1 ⊂ S)) both satisfy the right Ore condition, then also (R, T(R ⊂ S)) satisfies the right Ore condition and
RT(R ⊂ S)−1 ∼
= (RT −1 )(T(RT −1 ⊂ S))−1 . Both rings can be embedded into S and hence
R ⊂ (RT −1 )(T(RT −1 ⊂ S))−1 = RT(R ⊂ S)−1 ⊂ S.
(iv) If moreover T(RT −1 ⊂ S) = NZD(RT −1 ), then T(R ⊂ S) = NZD(R)
and RT(R ⊂ S)−1 is a classical ring of right fractions.
Proof. (i) If R is division closed in S, then D(R ⊂ S) = R. Since D(R ⊂ S)
is by definition division closed in S the claim follows. Similarly for the rational
closure.
(iii) We start with (a, s) ∈ R × T(R ⊂ S) and have to show that a right fraction
s−1 a can be written as a left fraction. Since R×T(R ⊂ S) ⊂ RT −1 ×T(RT −1 ⊂
S) and the assumption yields elements (b, t) ∈ R × T and (c, u) ∈ R × T with
cu−1 ∈ T(RT −1 ⊂ S) such that (symbolically)
s−1 a = (bt−1 )(cu−1 )−1 = b(cu−1 t)−1 .
Note that cu−1 ∈ T(RT −1 ⊂ S) implies c ∈ T(R ⊂ S) and therefore (b, cu−1 t) ∈
R × T(R ⊂ S). This is turned into a proof by avoiding inverses which not yet
exist. Once RT(R ⊂ S)−1 exists we know that R → RT(R ⊂ S)−1 is universal
T(R ⊂ S)-inverting. Similarly for R → RT −1 and RT −1 → (RT −1 )T(RT −1 ⊂
S)−1 . Now apply (ii).
(iv) It is sufficient to check NZD(R) ⊂ T(R ⊂ S). Let r ∈ NZD(R) be a
non-zerodivisor in R. Using the Ore condition for RT −1 one verifies that r ∈
NZD(RT −1 ) = T(RT −1 ⊂ S). So r becomes invertible in S.
59
For the proof of 8.7 (ii) we need some preparations. Two matrices (or elements)
a, b ∈ M(R) are called stably associated over R if there exist matrices c,
d ∈ GL(R), such that
a 0
b 0
c
d−1 =
0 1n
0 1m
with suitable m and n. Note that this is an equivalence relation and this relation
respects the property of being invertible. Given a set of matrices Σ ⊂ M(R) we
denote by Σ̃ the set of all matrices which are stably associated to a matrix in
Σ. The first statement of the following lemma is also known as Cramer’s rule.
Lemma 8.8. (i) Let Σ ⊂ M(R) be a set of matrices and f : R → RΣ be
universal Σ-inverting. Every matrix a ∈ M(RΣ ) is stably associated over
RΣ to a matrix f (b) with b ∈ M(R).
(ii) Let Σ = f −1 (GL(RΣ )) be the so called saturation of Σ, then f : R → RΣ
is universal Σ-inverting and similarly with Σ̃ instead of Σ. For purposes
of the universal localization we can always replace Σ by Σ̃ or Σ.
(iii) Let Σ, Σ0 ⊂ M (R). Let f : R → RΣ be universal Σ-inverting and let
RΣ → (RΣ )f (Σ0 ) be universal f (Σ0 )-inverting, then the composition g ◦ f :
R → (RΣ )f (Σ0 ) is universal Σ ∪ Σ0 -inverting and therefore
(RΣ )f (Σ0 ) ∼
= RΣ∪Σ0 .
Proof. (i) This can be deduced from a generator and relation construction of
RΣ . Compare [78, page 53].
(ii) By definition of Σ the map f : R → RΣ is Σ-inverting. One verifies that it
also has the corresponding universal property. Similarly for Σ̃.
(iii) Suppose h : R → S is Σ ∪ Σ0 -inverting. The universal property of f : R →
RΣ gives a unique map φ : RΣ → S which is f (Σ0 )-inverting. The universal
property of g : RΣ → (RΣ )f (Σ0 ) yields the desired unique Φ : (RΣ )f (Σ0 ) → S.
Now we are prepared to complete the proof of Proposition 8.7.
Proof of Proposition 8.7(ii). Let a be a matrix in Σ(RΣ → S). From 8.8 we
know that a (as every matrix over RΣ ) is stably associated to a matrix f (b)
with b ∈ R. For purposes of the universal localization we can therefore replace
Σ(RΣ → S) by f (Σ0 ) for a suitable set of matrices Σ0 ⊂ R. Compare 8.8. Now
by 8.8 we get
(RΣ )Σ(RΣ →S) ∼
= RΣ∪Σ0 .
= (RΣ )f (Σ0 ) ∼
One verifies that Σ ∪ Σ0 ⊂ Σ(R → S), i.e R → RΣ(R→S) is Σ ∪ Σ0 -inverting. An
application of 8.9 yields RΣ∪Σ0 ∼
= RΣ(R→S) .
The next lemma tells us that in some cases the different localizations coincide.
Lemma 8.9. Let R ⊂ S be a ring extension.
60
(i) Let Σ ⊂ Σ0 and R → RΣ be universal Σ-inverting. If R → RΣ is Σ0 inverting, then it is universal Σ0 -inverting.
(ii) Suppose R ⊂ D(R ⊂ S) is universal T(R ⊂ S)-inverting and von Neumann regular, then D(R ⊂ S) = R(R ⊂ S) and R ⊂ R(R ⊂ S) is
universal Σ(R ⊂ S)-inverting.
Proof. (i) Note that f has the universal property.
(ii) A von Neumann regular ring is division closed and rationally closed in
every overring, compare 13.15. Therefore D(R ⊂ S) = R(R ⊂ S). Since
R → R(R ⊂ S) = D(R ⊂ S) is Σ(R ⊂ S)-inverting an application of (i) with
Σ = T (R ⊂ S) and Σ0 = Σ(R ⊂ S) yields the result.
Now we apply the above to our situation. The second statement of the following
proposition is the abstract set-up for the most difficult part of the induction.
The last statement will be applied later only in the case where Γ is elementary
amenable.
Proposition 8.10. Let 1 → G → Γ → H → 1 be an exact sequence of groups.
(i) We have
and
D(DG ∗ H ⊂ UΓ)
R(RG ∗ H ⊂ UΓ)
= DΓ,
= RΓ.
(ii) Suppose CG → RG is universal Σ(G)-inverting and the pair
(RG ∗ H, T(RG ∗ H ⊂ UΓ))
fulfills the Ore condition. Then
D(RG ∗ H ⊂ UΓ) ∼
= (RG ∗ H)T(RG ∗ H ⊂ UΓ)−1 .
If this ring is von Neumann regular it coincides with RΓ, and in this case
CΓ ⊂ RΓ is universal Σ(Γ)-inverting.
(iii) If CΓ ⊂ DΓ is universal T(Γ)-inverting and DΓ is von Neumann regular,
then DΓ = RΓ and CΓ ⊂ DΓ is also universal Σ(Γ)-inverting.
(iv) Suppose (CG, T(G)) satisfies the Ore condition and one can show that
(DG ∗ H, T(DG ∗ H ⊂ UΓ)) satisfies the Ore condition, then
DΓ ∼
= (CΓ)T(Γ)−1
is an Ore localization and universal T(Γ)-inverting. If moreover T(DG ∗
H ⊂ UΓ) = NZD(DG ∗ H) then T(Γ) = NZD(CΓ).
61
Proof. (i) Since UΓ is von Neumann regular it is division closed, and from the
definition of the division closure we get DG = D(CG ⊂ UG) = D(CG ⊂ UΓ) ⊂
D(CΓ ⊂ UΓ) = DΓ. Since D ∗ G is the subring generated by DG and H we
have D ∗ G ⊂ DΓ. Now the result follows from 8.7 (i).
(ii) From 8.6 (iii) and the assumption we know that f : CG ∗ H ⊂ RG ∗ H
is universal Σ(G)-inverting. As always the Ore localization RG ∗ H → (RG ∗
H)T(RG ∗ H ⊂ UΓ)−1 is universal T(RG ∗ H ⊂ UΓ)-inverting and the map to
D(RG ∗ H ⊂ UΓ) given by the universal property is an isomorphism, compare
13.17. If this ring is von Neumann regular we can apply 8.9 (ii) and see that
g : RG ∗ H → R(RG ∗ H ⊂ UΓ) = D(RG ∗ H ⊂ UΓ)
is universal Σ(RG ∗ H ⊂ UΓ)-inverting. Now g ◦ f is an iterated localization as
in 8.7 (ii) and the result follows.
(iii) This is 8.9 (ii).
(iv) Apply 8.7 (iii) with R = CG ∗ H, S = UΓ and T = T(G). The second
statement is 8.7 (iv).
Remember that we want to prove not only von Neumann regularity but semisimplicity for the rings DΓ. The following criterion tells us that it is sufficient to
check the ascending chain condition. Note that taking division and rational
closures in our situation always leads to ∗-closed subrings of UΓ.
Lemma 8.11 (∗-Closed Subrings of UΓ). Let Γ be any group.
(i) Every ∗-closed subring of UΓ is semiprime.
(ii) A ring which is semiprime and artinian is semisimple.
Proof. (i) Let a be an operator in UΓ. Since a is densely defined and closed, the
closure of the graph of a|dom(a∗ a) is the graph of a. Now for x ∈ dom(a∗ a) ⊂
dom(a) we have < a∗ ax, x >= |ax|2 and therefore a∗ a = 0 implies a = 0. Now
given an ideal I 6= 0 we have to show that I 2 6= 0. Let a ∈ I be nontrivial, then
0 6= a∗ a ∈ I and 0 6= (a∗ a)∗ (a∗ a) = (a∗ a)2 ∈ I 2 .
(ii) See Theorem 2.3.10 on page 169 in [76].
Lemma 8.12 (Crossed Products and Chain Conditions). Let R ∗ G be a
crossed product.
(i) If R is artinian and if G is finite, then R ∗ G is artinian.
(ii) If R is noetherian and if G is finite,infinite cyclic or more generally
polycyclic-by-finite, then R ∗ G is noetherian. If R is noetherian and if
G∼
= Z is infinite cyclic, then R ∗ G is noetherian.
(iii) If R is semisimple of characteristic 0 and if G is arbitrary, then R ∗ G is
semiprime.
62
(iv) If R is semisimple of characteristic 0 and G finite, then R ∗ G is semisimple.
Proof. If G is finite, a chain of ideals in R ∗ G can be considered as a chain of Rmodules in the finitely generated R-module R ∗ G which is artinian respectively
noetherian as an R-module. This proves (i) and (ii) if G is finite.
(ii) Let G = Z be infinite cyclic. Let µ : Z → R ∗ G be the crossed product
˜ = (µ(1))n . This defines a new crossed
structure map, compare 14.1. Set µ(n)
product structure on R ∗ Z which has the advantage that µ̃ and therefore n 7→
cµ̃(n) is a group homomorphism and the corresponding twisting τ̃ is trivial. With
this new crossed product structure R∗Z is a skew-Laurent polynomial ring. Now
we first treat the corresponding skew-polynomial ring with the non-commutative
analogue of the Hilbert basis theorem ([76, Prop.3.5.2 on page 395]). The skewLaurent-polynomial ring is then an Ore localization of the skew polynomial ring
and therefore again noetherian by [76, Prop. 3.1.13. on page 354]. A polycyclicby-finite group (by definition, see [77][page 310]) can be obtained by iterated
extensions by infinite cyclic groups followed by an extension by a finite group.
Therefore an iterated use of the above arguments finishes the proof.
(iii) This is a particular case of Theorem I in [69] since a semisimple ring contains
no nilpotent ideals.
(iv) Combine (i), (ii) and 8.11(ii).
Now we are ready to prove
AIIa)
(A)Y ⇒ (A)Y{finite}
CIIa)
(C)Y ⇒ (C)Y{finite} .
Or in words:
Proposition 8.13 ( AIIa) and CIIa) ). Let
1→G→Γ→H →1
be an exact sequence of groups with H finite.
If DG is semisimple and universal Σ(G)-inverting, then
DΓ = DG ∗ H
is semisimple and universal Σ(Γ)-inverting.
If moreover T(G) = NZD(G) and if the pair (CG, T(G)) satisfies the Ore condition, then T(Γ) = NZD(Γ), the pair (CΓ, T(Γ)) satisfies the Ore condition and
DΓ is a classical ring of fractions for CΓ.
Proof. From the last lemma we know that DG∗H is semisimple and in particular
von Neumann regular and division closed (compare 13.15), hence DG ∗ H =
D(DG∗H ⊂ UΓ) = DΓ by 8.10 (i) and moreover T(DG∗H ⊂ UΓ) = (DG∗H)× .
Again, by semisimplicity DG = RG and DΓ = RΓ, and an application of 8.10
(ii) with T(DG ∗ H ⊂ UΓ) = (DG ∗ H)× (nothing needs to be inverted) yields
the result. Similarly the second statement follows from 8.10 (iv).
63
We now turn to the case where H is infinite cyclic. The crossed products which
occur in this case are particularly simple, they are so-called skew-Laurent polynomial rings. From Lemma 8.12 about crossed products and chain conditions
we know that DG ∗ Z is noetherian, ∗-closed and therefore semiprime 8.11(i).
The following theorem is therefore very promising.
Theorem 8.14 (Goldie’s Theorem). Let R be a right noetherian semiprime
ring. Let T = NZD(R) be the set of all non-zerodivisors in R, then (R, T )
satisfies the right Ore condition and the ring RT −1 is semisimple.
Proof. See section 9.4 in [16].
The crux of the matter is that we do not know that all non-zerodivisors become
invertible in UΓ. A priori we only have T(DG ∗ Z ⊂ UΓ) ⊂ NZD(DG ∗ Z).
Therefore even though we know that the Ore localization
DG ∗ Z → (DG ∗ Z)NZD(DG ∗ Z)−1
exists, is semisimple, and is as always universal NZD(DG ∗ Z)-inverting, it is not
clear that there is a map from this localization to UΓ. Our next aim is therefore
to prove the equality T(DG ∗ Z ⊂ UΓ) = NZD(DG ∗ Z).
At this point we have to use more than abstract ring theoretical arguments and
investigate the situation again in terms of functional analysis. The key point is
the following theorem of Linnell.
Theorem 8.15. Let 1 → G → Γ → Z → 1 be an exact sequence of groups.
Choose a section µ : Z → Γ which is a group homomorphism. Set z = µ(1).
Suppose the first non-vanishing coefficient of the Laurent polynomial f (z) ∈
N G ∗ Z is a non-zerodivisor in N G, then f (z) is a non-zerodivisor in N Γ.
Proof. Theorem 4 in [47] proves a more general statement.
This enables us to prove that non-zerodivisors in UG ∗ Z of a very special kind
become invertible in UΓ.
Corollary 8.16. Suppose f (z) ∈ UG ∗ Z is of the form f (z) = 1 + a1 z + . . . +
an z n , then it is invertible in UΓ.
Proof. Remember that UG is a left and a right ring of fractions of N G (Proposition 2.8). Since there is a common denominator for the a1 , . . . , an we can
find an s ∈ T(N G ⊂ UG) = NZD(N G) and a g(z) ∈ N G ∗ Z such that
f (z) = s−1 g(z). The first non-vanishing coefficient of g(z) is s and the above
theorem applies. We get g(z) ∈ NZD(N Γ) = T(N Γ ⊂ UΓ) ⊂ UΓ× and therefore f (z) = s−1 g(z) ∈ UΓ× .
Now we have to use the structure theory of semisimple rings to deduce the
desired result. This is done in several steps.
Proposition 8.17. Suppose R ∗ Z ⊂ S is a crossed product and one knows that
polynomials of the form f (z) = 1 + a1 z + . . . + an z n ∈ R ∗ Z become invertible
in S. Then if
64
(i) R is a skew field,
(ii) R is a simple artinian ring,
(iii) R is a semisimple ring,
then we have NZD(R ∗ Z) = T(R ∗ Z ⊂ S), and the Ore localization
(R ∗ Z)NZD(R ∗ Z)−1
embeds into S as the division closure
D(R ∗ Z ⊂ S).
Proof. We have already discussed above in 8.12 that R ∗ Z with R semisimple
is noetherian and semiprime. So indeed Goldie’s theorem applies, and once
we know that every non-zerodivisor becomes invertible in S the Ore localization
embeds into S as the division closure by Proposition 13.17(ii). In fact, with little
effort one could improve the proof given below and reprove Goldie’s theorem in
this special case.
(i) One verifies that up to a unit in R and multiplication by z n every nonzero
element of R ∗ Z can be written as f (z) above. The Ore localization in this case
is a skew field.
(ii) Step1: We will first change the crossed product structure on the ring R∗Z to
obtain one with better properties. Let R = Mn (Z) ⊗Z D with D a skew field be
a simple artinian ring. Let µ denote the crossed product structure map for the
ring R ∗ Z. Every automorphism c : R → R splits into an inner automorphism
and an automorphism of the form
id ⊗ θ : Mn (Z) ⊗Z D → Mn (Z) ⊗Z D,
where θ is an automorphism of D, compare [38, page 237]. In particular cµ(1) =
cu ◦ (id ⊗ θ) for some invertible element u ∈ R = Mn (Z) ⊗Z D. If we now set
µ̃(1) = u−1 µ(1) and µ̃(n) = (µ̃(1))n we get a new crossed product structure
map µ̃ such that
cµ̃(n) : Mn (Z) ⊗Z D → Mn (Z) ⊗Z D
restricts to
cµ̃(n) : Z · 1n ⊗Z D → Z · 1n ⊗Z D
and the corresponding twisting τ̃ is identically 1. We see that with respect to
this new crossed product structure the subring (or left Z · 1n ⊗Z D-module)
generated by Z · 1n ⊗Z D and µ̃(Z) is a sub-crossed product of (R ∗ Z, µ̃), i.e.
the inclusion is a crossed product homomorphism. We denote this ring by
((Z · 1n ⊗Z D) ∗ Z, µ̃). Note that the assumption about polynomials is invariant
under this base change since µ(n) and µ̃(n) differ by an invertible element in R.
There is a natural isomorphism
Mn (Z) ⊗Z ((Z · 1n ⊗Z D) ∗ Z) → (Mn (Z) ⊗Z D) ∗ Z.
65
From now on we will write D instead of Z · 1n ⊗Z D, and via the above isomorphism we consider
Mn (Z) ⊗Z (D ∗ Z)
⊂
S
as a subring of S.
Step2: If we apply (i) to the inclusion Z · 1n ⊗Z (D ∗ Z) ⊂ S we see that
Z · 1n ⊗Z (D ∗ Z) fulfills the Ore condition with respect to the set T of all
nonzero elements. The Ore localization is a skew field K and embeds into S as
the division closure
K = (Z · 1n ⊗Z (D ∗ Z)) T −1 = D (Z · 1n ⊗Z (D ∗ Z) ⊂ S)
⊂
S.
Since K is an Ore localization the following diagram is a push-out diagram by
Corollary 13.8.
Z · 1n ⊗Z (D ∗ Z)
- Z · 1n ⊗Z K
?
?
Mn (Z) ⊗Z (D ∗ Z) - Mn (Z) ⊗Z K.
Since Z · 1n ⊗Z K embeds into S and Mn (Z) ⊗Z (D ∗ Z) embeds into S there is
an induced homomorphism
Φ : Mn (Z) ⊗Z K → S.
Since D(Mn (Z)⊗Z (D∗Z) ⊂ S) contains Mn (Z)⊗Z (D∗Z) and Z·1n ⊗Z K it must
contain im(Φ). Since im(Φ) is a homomorphic image of a the semisimple ring
Mn (Z) ⊗Z K it is semisimple and therefore division closed. We get equality. We
claim that Φ is injective. The entries of a matrix in Mn (Z) ⊗Z K can always be
brought to a common denominator. Injectivity follows from the above pushout
diagram and the fact that the Mn (Z) ⊗Z (D ∗ Z) → S is injective. So we have
that
Mn (Z) ⊗Z K ∼
= D (Mn (Z) ⊗Z (D ∗ Z) ⊂ S)
⊂
S.
Remains to be proven that all non-zerodivisors in Mn (Z) ⊗Z (D ∗ Z) become
invertible in S. Let a be an element in Mn (Z) ⊗ Z(D ∗ Z). Now a either
becomes a zerodivisor or invertible in the semisimple ring Mn (Z) ⊗Z K. From
Proposition 13.7 we know that
Mn (Z) ⊗Z K ∼
= (Mn (Z) ⊗Z (D ∗ Z)) (T · 1n )−1
is an Ore localization. Using this we verify that a must have been a zerodivisor
from the beginning
if it becomes a zerodivisor in Mn (Z) ⊗Z K.
P
(iii) Let 1 =
ei be a decomposition of the unit in R into central primitive
orthogonal idempotents. Every automorphism of R permutes the ei . Therefore
66
we can find an m ∈ N with cm
µ(1) (ei ) = ei , and the product structure induced
on the subring R ∗ (mZ) ⊂ R ∗ Z respects the decomposition of R into simple
artinian rings, i.e. the ei are still central orthogonal idempotents for R ∗ mZ
R ∗ mZ = (⊕i Rei ) ∗ mZ = ⊕i (R ∗ mZ)ei .
The hypothesis implies that every f (z) ∈ (R ∗ mZ)ei with leading term ei is
invertible in ei Sei . From (ii) we therefore get
NZD(R ∗ mZei ) = T(R ∗ mZei ⊂ ei Sei ).
One checks that an element f ∈ R ∗ mZ is invertible in ei Sei if and only if every
f ei is invertible in ei Sei , and it is a non-zerodivisor in R ∗ mZ if and only if all
f ei are non-zerodivisors in R ∗ mZei . Therefore
NZD(R ∗ mZ) = T(R ∗ mZ ⊂ S).
Goldie’s Theorem yields that
D(R ∗ mZ ⊂ S) = (R ∗ mZ)T(R ∗ mZ ⊂ S)−1
is semisimple. Note that R ∗ Z = (R ∗ mZ) ∗ Z/mZ and D(R ∗ mZ ⊂ S) ∗ Z/mZ
is semisimple by 8.12 (iv) and therefore division closed. So T(R ∗ Z ⊂ S) =
T(R ∗ Z ⊂ D(R ∗ mZ ⊂ S) ∗ Z/mZ). By 8.5 (vi) we have that
(R ∗ mZ) ∗ Z/mZ ⊂
=
=
D(R ∗ mZ ⊂ S) ∗ Z/mZ
(R ∗ mZ)T(R ∗ mZ ⊂ S)−1 ∗ Z/mZ
((R ∗ mZ) ∗ Z/mZ)T(R ∗ mZ ⊂ S)−1
is universal T(R ∗ mZ ⊂ S)-inverting. Since it is also T(R ∗ Z ⊂ S)-inverting
it is universal T(R ∗ Z ⊂ S)-inverting because one easily checks the corresponding universal property, compare 8.9(i). Therefore it is isomorphic to the Ore
localization given by Goldie’s theorem.
It would be very interesting to have a similar statement if R is von Neumann
regular, but of course the proof given here relies heavily on the structure theory
for semisimple rings.
Combining the above results we get
AIIb)
(A)Y ⇒ (A)Y{infinite cyclic}
CIIb)
(C)Y ⇒ (C)Y{infinite cyclic} .
Or in words:
Proposition 8.18 ( AIIb) and CIIb) ). Let 1 → G → Γ → Z → 1 be an
exact sequence of groups. If DG is semisimple and if CG ⊂ DG is universal
Σ(G)-inverting, then
DΓ = (DG ∗ Z)T(DG ∗ Z ⊂ UΓ)−1
67
is semisimple and CΓ ⊂ DΓ is universal Σ(Γ)-inverting.
If moreover T(G) = NZD(CG) and if DG ∼
= (CG)T(G)−1 is a classical ring of
fractions for G, then T(Γ) = NZD(CΓ) and
DΓ ∼
= (CΓ)T(Γ)−1
is a classical ring of quotients for CΓ.
Proof. From the above discussion we know that DG ∗ Z is semiprime noetherian
and T(DG ∗ Z ⊂ UΓ) = NZD(DG ∗ Z). Therefore the Ore localization
(DG ∗ Z)T(DG ∗ Z ⊂ UΓ)−1
exists, is semisimple according to Goldie’s Theorem and isomorphic to the division closure D(DG ∗ Z ⊂ UΓ) = DΓ. That CΓ ⊂ DΓ is universal Σ(Γ)-inverting
now follows from 8.10 (ii), where we identify DG and DΓ with RG and RΓ by
8.10 (iii). The last statement is 8.10 (iv).
One extension by a finitely generated abelian-by-finite group can always be
replaced by finitely many extensions by an infinite cyclic group followed by one
extension by a finite group. If we now combine AIIa) and AIIb) and similarly
CIIa) and CIIb) we immediately get the following:
AIIa)+b)
(A)Y ⇒ (A)Y{f.g. abelian by finite}
CIIa)+b)
(C)Y ⇒ (C)Y{f.g. abelian by finite} .
The statement (A) only cares about the ring extension CΓ ⊂ DΓ. But we have
also obtained information about the intermediate extension DG∗H ⊂ DΓ which
we will need again in the (B)-part. We record a corresponding statement in the
following Lemma.
Lemma 8.19. Let 1 → G → Γ → H → 1 be an exact sequence of groups with
H finitely generated abelian-by-finite.
If DG is semisimple, then
T(DG ∗ H ⊂ UΓ) = NZD(DG ∗ H),
the pair (DG ∗ H, T(DG ∗ H ⊂ UΓ)) satisfies the Ore condition,
DΓ ∼
= (DG ∗ H)T(DG ∗ H ⊂ UΓ)−1
is a classical ring of fractions for DG ∗ H and semisimple.
Proof. In the proof of AIIb) we have already seen the corresponding statement
for H = Z. Since an extension by a finitely generated abelian-by-finite group
can be replaced by finitely many extensions by infinite cyclic groups followed
by an extension by a finite group it remains to be verified that we can iterate.
Apply 8.7 (iv) and 8.5 (vi).
68
8.2
Induction Step: Extensions - The (B)-Part
We now turn to the (B)-part of the induction step. Remember that our task
is to prove the surjectivity of the map colimK∈F in(Γ) K0 (CK) → K0 (DΓ). If
we combine 8.19 with the following proposition we can handle the passage from
DG ∗ H to DΓ if H is finitely generated abelian-by-finite. Remember that for a
ring R the abelian group G0 (R) is the free abelian group on isomorphism classes
of finitely generated modules modulo the usual relations which come from exact
sequences.
Proposition 8.20. (i) Suppose that T ⊂ R is a set of non-zerodivisors and
that the pair (R, T ) satisfies the right Ore condition, then the map R →
RT −1 induces a surjection
G0 (R) → G0 (RT −1 ).
(ii) If the ring R is semisimple, then the natural map K0 (R) → G0 (R) is an
isomorphism.
Proof. (i) Let M be a finitely generated RT −1 -module. We have to find a
finitely generated R-module N with N ⊗R RT −1 ∼
= M . Choose an epimorphism
p : (RT −1 )n → M . Let i : Rn → (RT −1 )n be the natural map. Now N =
p ◦ i(Rn ) is a finitely generated R-submodule of M . Let j : N → M denote the
inclusion. Now apply − ⊗R RT −1 to the diagram
- N
Rn
j
i
?
(RT −1 )n
?
- M.
Since localizing is exact j ⊗ idRT −1 is injective. For every RT −1 -module L we
have that L ⊗R RT −1 ∼
= L. In particular i ⊗ idRT −1 is an isomorphism and
therefore also j⊗RT −1 is an isomorphism and N ⊗R RT −1 ∼
= M.
(ii) Over a semisimple ring all modules are projective.
Our target DΓ as well as our sources CK (where K is finite) are semisimple
rings, and thus we can pass to G-theory instead of K-theory during the proof.
Proposition 8.21. Suppose DG is semisimple and H is finitely generated abelianby-finite, then the map
G0 (DG ∗ H) → G0 (DΓ) = K0 (DΓ)
is surjective.
Proof. Combine 8.19 and 8.20.
69
Before we go on let us explain why the naive approach of proving BIIa) and
BIIb) separately would fail. In fact it is possible to prove a statement like
BIIb)
(A)Y and (B)Y ⇒ (B)Y{infinite cyclic} .
Given an extension 1 → G → Γ → Z → 1, every finite subgroup of Γ already
lies in the subgroup G, and we are left with the task of proving that the map
G0 (DG) → G0 (DG ∗ Z) is surjective. This is possible. Unfortunately extensions
by finite groups are much more complicated. The map G0 (DG) → G0 (DG∗H) is
of course in general not surjective if H is finite and G0 (DG ∗ H) really depends
on the crossed product structure (take for example the trivial group for G).
The ring theoretical structure of DG and the abstract knowledge of H is not
sufficient. Fortunately we have the following theorem.
Theorem 8.22 (Moody’s Induction Theorem). Let R be a noetherian ring
and H be a group which is a finite extension of a finitely generated abelian group.
Then the natural map
colimK∈F in(H) G0 (R ∗ K) → G0 (R ∗ H)
is surjective.
Proof. This is due to Moody [60] and [61]. See also [13], [26] and Chapter 8 of
[70].
In our situation we get in particular.
Proposition 8.23. Let 1 → G → Γ → H → 1 be an exact sequence with H
finitely generated abelian-by-finite. Suppose DG is semisimple, then the map
colimL∈F in(H) G0 (DG ∗ L) → G0 (DG ∗ H)
is surjective.
Combining this with the induction hypothesis we finally get:
BIIa)+b)
Y{finite} = Y, (A)Y and (B)Y ⇒ (B)Y{f.g. abelian-by-finite} .
In words:
Proposition 8.24 ( BIIa)+b) ). Let Y be a class of groups which is closed
under finite extensions. Let 1 → G → Γ → H → 1 be an exact sequence
with G ∈ Y and let H be finitely generated abelian-by-finite. Suppose DG is
semisimple and we know that
colimL∈F inG K0 (CL) → K0 (DG)
is surjective for all groups G in Y. Then the map
colimK∈F inΓ K0 (CK) → K0 (DΓ)
is surjective.
70
Proof. Let p : Γ → H be the quotient map. All maps in the following diagram
are either isomorphisms or surjective.
∼
=
colimM∈F inΓ K0 (CM ) → colimK∈F inH colimL∈F inp−1 (K) K0 (CL)
→ colimK∈F inH K0 (Dp
∼
=
→ colimK∈F inH G0 (Dp
−1
−1
(1)
(K))
(2)
(K))
(3)
∼
=
→ colimK∈F inH G0 (DG ∗ K)
(4)
→ G0 (DG ∗ H)
→ G0 (DΓ)
(5)
(6)
→ K0 (DΓ)
(7)
∼
=
We start at the bottom. (7) is an isomorphism by 8.20 (ii) since we know from
AIIa)+b) that DΓ is semisimple. Map (6) is surjective by 8.21, map (5) by
Moody’s induction theorem. (4) Since K is finite and DG semisimple we know
from AIIa) that Dp−1 (K) = DG ∗ K. (3) This follows from 8.20 (ii). (2) By
assumption the class Y is closed under extension by finite groups. Therefore
p−1 (K) is again in Y and we know that
colimL∈F inp−1 (K) K0 (CL) → K0 (Dp−1 (K))
is surjective. Since every M ∈ FinΓ occurs as M ∈ Fin(p−1 (p(M ))) with
p(M ) ∈ FinH we get (1).
8.3
Induction Step: Directed Unions
This step is much easier than the preceding one. But the extra hypothesis about
the orders of finite subgroups enters the proof. First we collect what we have
to know about directed unions and localizations.
Proposition 8.25 (Directed Unions
Let Ri ⊂ Si ⊂ S
S and Localization).
S
be subrings for all i ∈ I. Let R = i∈I Ri and i∈I Si be directed unions.
Suppose all the rings Si are von Neumann regular.
(i) A directed union of von Neumann regular rings is von Neumann regular.
(ii) We have
T(R ⊂ S) =
Σ(R ⊂ S) =
and the unions are directed.
71
[
T(Ri ⊂ Si )
[
Σ(Ri ⊂ Si ),
i∈I
i∈I
(iii) We have
D(R ⊂ S) =
R(R ⊂ S) =
[
D(Ri ⊂ Si )
[
R(Ri ⊂ Si ),
i∈I
i∈I
and the unions are directed.
(iv) If all the D(Ri ⊂ Si ) are universal T(Ri ⊂ Si )-inverting, then D(R ⊂ S)
is universal T(S)-inverting.
(v) If all the R(Ri ⊂ Si ) are universal Σ(Ri ⊂ Si )-inverting, then R(R ⊂ S)
is universal Σ(S)-inverting.
(vi) Suppose all the pairs (Ri , T(Ri ⊂ Si )) satisfy the right Ore condition, then
(R, T(R ⊂ S)) satisfies the right Ore condition.
(vii) If NZD(Ri ) = T(Ri ⊂ Si ) for all i ∈ I, then NZD(R) = T(R ⊂ S).
(viii) If D(Ri ⊂ Si ) is a classical ring of right quotients for Ri , then D(R ⊂ S)
is a classical ring of right quotients for R.
Proof. (i) This follows immediately if one makes use of the fact that a ring R
is von Neumann regular if and only if for every x ∈ R there exists a y ∈ R such
that xyx = x.
(ii) Suppose x ∈ R becomes invertible in S, then since in the von Neumann
regular ring Si an element is either invertible or a zerodivisor (see 12.3(i)) it
is already invertible in Si . The other inclusion is clear. The argument for
Σ(R ⊂ S) is similar using that matrix rings over von Neumann regular rings
are again von Neumann regular.
(iii) Since Si is von Neumann
regular we have D(Ri ⊂ Si ) = D(Ri ⊂ S) ⊂
S
D(R
⊂
S)
and
therefore
D(R
i ⊂ Si ) ⊂ D(R ⊂ S). On the other hand
i∈I
S
S
D(R
⊂
S) is division closed in S. Similar for the
D(R
⊂
S
)
=
i
i
i
i∈I
i∈I
rational closure.
(iv) Given a T(R ⊂ S)-inverting homomorphism R → R0 . The composition
Ri → R → R0 is T(Ri ⊂ Si )-inverting and the universal property gives a unique
map φi : D(Ri ⊂ Si ) → R0 . Using that the union is directed and each φi
is
S unique one verifies 0that these maps determine a unique map D(R ⊂ S) =
i∈I D(Ri ⊂ Si ) → R .
(v) is similar to (iv).
(viii) follows from (vi) and (vii) which are easy.
If we apply this to our situation we get:
S
Proposition 8.26. If Γ = i∈I Γi is a directed union of groups the following
holds.
(i) The ring DΓ is the directed union of the subrings DΓi and RΓ is the
directed union of the RΓi .
72
(ii) Suppose CΓi ⊂ DΓi is universal T(Γi )-inverting for every i ∈ I. Then
CΓ ⊂ DΓ is universal T(Γ)-inverting.
(iii) Suppose CΓi ⊂ RΓi is universal Σ(Γi )-inverting for every i ∈ I. Then
CΓ ⊂ RΓ is universal Σ(Γ)-inverting.
(iv) Suppose for all i ∈ I that NZD(CΓi ) = T(Γi ) and suppose DΓi is a
classical ring of quotients for CΓi , then NZD(CΓ) = T(Γ) and DΓ is
a classical ring of fractions for CΓ.
(v) If all the DΓi are von Neumann regular, then DΓ is von Neumann regular.
(vi) If all the RΓi are von Neumann regular, then RΓ is von Neumann regular.
We now start with the (B)-part.
(B)Y ⇒ (B)LY .
BIII)
Somewhat more precisely:
Proposition 8.27 (BIII)). Let Γ be the directed union of the Γi , and suppose
that all the maps colimK∈F in(Γi ) K0 (CK) → K0 (DΓi ) are surjective. Then also
the map
colimK∈F in(Γ) K0 (CK) → K0 (DΓ)
is surjective.
Proof. From the preceding proposition we know that DΓ is the directed union
of the DΓi . Since K-theory is compatible with colimits and K ∈ FinΓ lies in
some F inΓi the result follows.
We now turn to the (A)-part. In our situation the DΓi are semisimple and
hence von Neumann regular. Note that the above proposition only yields that
DΓ is von Neumann regular. We have already seen in example 5.11 that without
the assumption on the orders of finite subgroups semisimplicity need not hold.
But since we have already established the (B)-part we can apply 5.12 to get
AIII)
(A)Y 0 and (B)Y 0 ⇒ (A)(LY)0
CIII)
(C)Y 0 and (B)Y 0 ⇒ (C)(LY)0 .
In words:
Proposition 8.28 (AIII) and CIII)). Let Γ be the directed union of the Γi .
Suppose all the DΓi are semisimple and universal Σ(Γi )-inverting. Suppose all
the maps
colimK∈F in(Γi ) K0 (CK) → K0 (DΓi )
are surjective. If there is a bound on the orders of finite subgroups of Γ, then
DΓ is semisimple and universal Σ(Γ)-inverting.
If moreover NZD(CΓi ) = T(Γi ) and all the pairs (CΓi , T(Γi )) satisfy the Ore
condition, then NZD(CΓ) = T(Γ) and the pair (CΓ, T(CΓ)) satisfies the Ore
condition.
73
Proof. The first part has been explained above. The second part is 8.26 (iv).
8.4
Starting the Induction - Free Groups
We begin with the promised proof of Theorem 5.16. Because of the following we
need not distinguish between the rational and the division closure in the case
of a torsionfree group.
Note 8.29. The ring DΓ is a skew field if and only if RΓ is a skew field, and
in this case the two rings coincide.
Proof. Suppose DΓ is a skew field. Then it is not only division closed but also
rationally closed by 13.15 and therefore DΓ = RΓ is a skew field. Conversely,
if RΓ is a skew field, then DΓ is a division closed subring of a skew field and
therefore itself a skew field.
Proposition 8.30. Let Γ be a torsionfree group and let R ⊂ C be a subring.
The Atiyah conjecture with R-coefficients holds if and only if D(RΓ ⊂ UΓ) is a
skew field.
Proof. Suppose DΓ = D(RΓ ⊂ UΓ) is a skew field. Let M be a finitely presented RΓ-module, then M ⊗RΓ DΓ is a finite dimensional vector space and
dimU Γ (M ⊗RΓ DΓ ⊗DΓ UΓ) is an integer.
On the other hand suppose the Atiyah conjecture holds. Let x be a nontrivial
element in the rational closure RΓ = R(RΓ ⊂ UΓ). We will show that x is
invertible in UΓ. Since RΓ is rationally closed and in particular division closed
the inverse x−1 lies in RΓ and we see that this ring is a skew field. By the note
above also DΓ is a skew field. Let 0 6= x ∈ RΓ. The following lemma tells us
that x is stably associated over RΓ (over UΓ would be sufficient) to a matrix A
over RΓ. InterpretingL
all matrices as matrices over UΓ we get an isomorphism of
n
UΓ-modules im(x) ⊕ i=1 UΓ ∼
= im(A). By the Atiyah conjecture dim(im(A))
has to be an integer and therefore dim im(x) is either 0 or 1. Additivity of
the dimension and von Neumann regularity of UΓ implies that x is either 0 or
invertible.
Lemma 8.31. Let R ⊂ S be a ring extension. Then every matrix over R(R ⊂
S) is stably associated over R(R ⊂ S) to a matrix over R.
Proof. We deduce this from the corresponding statement for the universal localization. Let R → RΣ(R⊂S) be universal Σ(R ⊂ S)-inverting. We know that the
map RΣ(R⊂S) → R(R ⊂ S) given by the universal property is surjective. So we
lift the matrix to RΣ(R⊂S) use 8.8 (i) and map everything back to R(R ⊂ S).
We have already seen that the Atiyah conjecture holds for the free group on two
generators. Since the conjecture is stable under taking subgroups we get from
the preceding proposition.
Proposition 8.32. If Γ is a free group, then DΓ is a skew field.
74
Proof.
The group Γ is the directed union of its finitely generated subgroups
S
i∈I Γi . Every finitely generated free group is a subgroup of the free group on
two generators. Since the Atiyah conjecture is stable under taking subgroups
the preceding
proposition yields, that all the DΓi are skew fields. By 8.26
S
DΓ = i∈I DΓi . One easily verifies that a directed union of skew fields is again
a skew field.
The hard part is now to verify that CΓ ⊂ DΓ is universal Σ(Γ)-inverting. Unfortunately we have to introduce a new concept: the universal field of fractions of
a given ring R. This should not be confused with any of the other notions introduced so far (universal Σ-inverting ring homomorphism, universal T -inverting
ring homomorphism, classical ring of fractions). As the name suggests, the universal field of fractions is defined by a universal property, but unfortunately it
does not have to exist. For more details the reader is referred to Appendix III.
Proposition 8.33. Let R be a semifir, then the universal field of fractions
R → K exists and is universal Σ(R → K)-inverting.
Proof. See Appendix III Proposition 13.20.
The group ring CΓ of a free group is the standard example of a fir [16][Section
10.9]. In particular it is a semifir. We still have to recognize DΓ as a universal
field of fractions for CΓ. From Corollary 13.25 we know that it is sufficient to
show that CΓ ⊂ DΓ is Hughes-free. By definition this means the following:
Given any finitely generated subgroup G ⊂ Γ and a t ∈ G together with a
homomorphism pt : G → Z which maps t to a generator, the set {ti | i ∈ Z}
is D(C ker pt ⊂ UΓ)-left linearly independent. But we know from the exact
sequence
1 → ker pt → G → Z → 1
and Proposition 8.5(i) that the ring generated by
D(C ker pt ⊂ UΓ) = D(C ker pt ⊂ U ker pt ) = D(ker pt )
and G as a subring of U(ker pt ) ∗ Z is itself a crossed product D(ker pt ) ∗ Z. In
particular it is a free D(ker pt )-module with basis {ti | i ∈ Z}.
Proposition 8.34 (AIa)). If Γ is a free group, then DΓ is a skew field and
CΓ ⊂ DΓ is universal Σ(Γ)-inverting.
Note that once we know DΓ is a skew field the (B)-part is trivially verified since
K0 (DΓ) ∼
= Z.
8.5
Starting the Induction - Finite Extensions of Free Groups
We now turn to the case of finite extensions of a free groups. We already know
from AIIa) that extensions by finite groups cause no problems for the (A)-part.
75
Proposition 8.35 (AIb)). Let Γ be a finite extension of a free group, then DΓ
is semisimple and CΓ ⊂ DΓ is universal Σ(Γ)-inverting.
The (B)-part remains to be proven. This time we really follow the philosophy
outlined in Subsection 5.3 and begin with the statement that would follow from
the isomorphism conjecture in algebraic K-theory.
Proposition 8.36. Let Γ be a finite extension of a free group, then the map
colimK∈F in(Γ) K0 (CK) → K0 (CΓ)
is surjective.
Proof. See [48][Lemma 4.8].
We still have to show surjectivity of the map K0 (CΓ) → K0 (DΓ). Since we
already know that CΓ ⊂ DΓ is universal Σ(Γ)-inverting we can use results of
Schofield [78] about the K-theory of universal localizations. Again we have to
introduce a few new concepts which we discuss in Appendix III.
Proposition 8.37. Let R be a hereditary ring with faithful projective rank function ρ : K0 (R) → R and let Σ be a set of full matrices with respect to ρ, then
the universal Σ-inverting homomorphism R → RΣ induces a surjection
K0 (R) → K0 (RΣ ).
Proof. This is a simplification of Theorem 5.2 of [78]. Note that by Lemma 1.1
and Theorem 1.11 in the same book the projective rank function is automatically
a Sylvester projective rank function and by Theorem 1.16 there are enough left
and right full maps.
Let us explain how this applies in our situation. Let Γ be finitely generated and
let G be a free subgroup of finite index. Then Γ operates with finite isotropy on
a tree, and the CΓ-module C admits of a 1-dimensional projective resolution.
Compare [21][Chapter IV, Corollary 3.16 on page 114]. This implies that every
CΓ-module has a 1-dimensional projective resolution (see [77][Proposition 8.2.19
on page 315]), and in particular CΓ is hereditary. Now the dimension function
for UΓ gives us a faithful projective rank function ρ = dimU Γ (− ⊗CΓ UΓ). We
check in Lemma 13.30 that Σ(CΓ ⊂ UΓ) is a set of full maps with respect to ρ.
Let us record the result.
Proposition 8.38. If Γ is a finite extension of a finitely generated free group,
then the map K0 (CΓ) → K0 (DΓ) is surjective.
Since K-theory is compatible with colimits and DΓ is the directed union of the
DΓi for the finitely generated subgroups Γi of Γ we finally get our result.
Proposition 8.39 (BIb)). If Γ is a finite extension of a free group, then the
map
colimK∈F in(Γ) K0 (CK) → K0 (DΓ)
is surjective.
This finishes the proof of Theorem 8.3 and Theorem 8.4.
76
9
Homological Properties and Applications
In this section we will apply our careful investigation of the rings DΓ to obtain
homological information about our rings. We have already mentioned that for
non-amenable groups we cannot expect that the functor − ⊗CΓ UΓ is exact. The
following gives a precise result for groups in the class C.
Theorem 9.1. (i) Let Γ be in the class C with a bound on the orders of finite
subgroups, then
TorCΓ
p (−; DΓ) = 0
for all p > 1.
(ii) If moreover Γ is elementary amenable, then
TorCΓ
p (−; DΓ) = 0
for all p > 0,
i.e. the functor − ⊗CΓ DΓ is exact.
Note that for these groups DΓ is semisimple and therefore the functor − ⊗DΓ
UΓ is exact. The functor − ⊗ZΓ CΓ is always exact. Therefore we immediZΓ
ately get the corresponding statements for TorCΓ
p (−; UΓ), Torp (−; DΓ) and
ZΓ
Torp (−; UΓ). This has consequences for L2 -homology.
Corollary 9.2. Let Γ be in C with a bound on the orders of finite subgroups.
Then there is a universal coefficient theorem for L2 -homology: Let X be a Γspace whose isotropy groups are all finite, then there is an exact sequence
0 → Hn (X; Z) ⊗ZΓ UΓ → HnΓ (X; UΓ) → Tor1 (Hn−1 (X; Z); UΓ) → 0.
Proof. If X has finite isotropy, then the set of singular simplices also has only
finite isotropy groups. If H is a finite subgroup of Γ, then C [Γ/H] ∼
= CΓ ⊗CH C
is induced from the projective CH-module C and therefore projective. We see
that the singular chain complex with complex coefficients C∗ = C∗sing (X; C)
is a complex of projective CΓ-modules. The E 2 -term of the Künneth spectral
sequence (compare Theorem 5.6.4 on page 143 in [85])
2
Epq
= TorCΓ
p (Hq (C∗ ); DΓ) ⇒ Hp+q (C∗ ⊗ DΓ) = Hp+q (X; DΓ)
is concentrated in two columns. The spectral sequence collapses, and we get
exact sequences
0 → Hn (X; C) ⊗CΓ DΓ → Hn (X; DΓ) → TorCΓ
1 (Hn−1 (X; C); DΓ) → 0.
Applying the exact functor − ⊗DΓ UΓ yields the result.
If we apply Theorem 9.1 to the trivial CΓ-module C we get:
77
Corollary 9.3. If Γ belongs to C and has a bound on the orders of finite subgroups, then
Hp (Γ; UΓ) = TorCΓ
p (C; UΓ) = 0
for all p ≥ 0.
(2)
In particular, if the group is infinite we have b0 (Γ) = 0 and therefore
χ(2) (Γ) ≤ 0,
(2)
since b1 (Γ) is the only L2 -Betti number which could possibly be nonzero.
As we have discussed in Section 5, the L(2) -Euler characteristic coincides with
the virtual or the usual Euler-characteristic whenever these are defined.
The proof of Theorem 9.1 depends on the following Lemma.
Lemma 9.4. (i) Let R ∗ G ⊂ S ∗ G be a compatible ring extension. Let M
be an R ∗ G-module. There is a natural isomorphism of right S-modules
R∗G
TorpR∗G (M ; S ∗ G) ∼
M ; S)
= TorR
p (resR
for all p ≥ 0.
S
(ii) Suppose R ⊂ S is a ring extension and R = i∈I Ri is the directed union
of the subrings Ri . Let M be an R-module. Then there is a natural
isomorphism of right S-modules
Ri
R
∼
TorR
p (M ; S) = colimi∈I Torp (resRi M ; Si ) ⊗Si S
for all p ≥ 0.
Proof. (i) We start with the case p = 0. As usual we denote the crossed product
structure map by µ. Define a map
R∗G
hM : resR
M ⊗R S → M ⊗R∗G S ∗ G
by m ⊗ s 7→ m ⊗ s. Obviously h is a natural transformation from the functor
R∗G
∼
(−)⊗R S to −⊗R∗G S ∗G. If M = R∗G the map h−1
resR
R∗G : R∗G⊗R∗G S ∗G =
R∗G
R ∗ G ⊗R S given by sµ(g) 7→ g ⊗ c−1
(s)
is
a
well-defined
inverse.
S ∗ G → resR
g
Since h is compatible with direct sums we see that hF is an isomorphism for
all free modules F . Now if M is an arbitrary module choose a free resolution
F∗ → M of M and apply both functors to
F1 → F0 → M → 0 → 0.
Both functors are right exact, therefore an application of the five lemma yields
the result for p = 0. Now let P∗ → M be a projective resolution of M , then
R∗G
TorR
M ; S)
p (resR
=
∼
=
R∗G
Hp (resR
P∗ ⊗R S)
→ Hp (P∗ ⊗R∗G S ∗ G)
=
78
TorR
p (M ; S ∗ G).
(ii) Again we start with the case p = 0. The natural surjections resR
Ri M ⊗Ri S →
M ⊗R S induce a surjective map
hM : colimi∈I resR
Ri M ⊗Ri S → M ⊗R S
which is natural in M . Suppose the element of the colimit represented by
P
R
k mk ⊗ sk ∈ resRi M ⊗Ri S is mapped to zero in M ⊗R S. By construction the
tensor product M ⊗R S is the quotient of the free module on the set M × S by a
relation submodule. But every relation
P involves only finitely many elements of
R, so we can find a j ∈ I such that k mk ⊗ sk = 0 already in resR
Rj M ⊗Rj S.
We see that hM is an isomorphism. Now let P∗ → M be a projective resolution.
Since the colimit is an exact functor it commutes with homology and we get
i
colimi∈I TorR
p (M ; S)
=
=
∼
=
colimi∈I Hp (resR
Ri P∗ ⊗Ri S)
Hp (colimi∈I (resR
Ri P∗ ⊗Ri S))
→ Hp (P∗ ⊗R S)
=
TorR
p (M ; S).
Proof of Theorem 9.1. (i) The proof works via transfinite induction over the
group as explained in 7.9. (I) We first verify the statement for free groups.
Let Γ be the free group generated by the set S. The cellular chain complex
of the universal covering of the obvious 1-dimensional classifying space gives a
projective resolution of the trivial module of length one
M
0→
CΓ → CΓ → C → 0.
S
Now if M is an arbitrary CΓ-module we apply − ⊗C M to the above complex
and get a projective resolution of length 1 for M (diagonal action), compare
Proposition 8.2.19 on page 315 in [77]. In particular we see that
TorCΓ
p (M ; DΓ) = 0
for p > 1.
(II)” The next step is to prove that the statement remains true under extensions
by finite groups. So let 1 → G → Γ → H → 1 be an exact sequence with H
finite. From Proposition 8.13 we know that DΓ = DG ∗ H. Let M be a CΓmodule, then with Lemma 9.4 and the induction hypothesis we conclude
TorCΓ
p (M ; DΓ) =
∼
=
=
TorpCG∗H (M ; DG ∗ H)
CG∗H
TorCG
M ; DG)
p (resCG
0 for p > 1.
The case H infinite cyclic is only slightly more complicated. This time we know
from Proposition 8.18 (AIIb)) that DΓ = (DG ∗ H)T −1 is an Ore localization.
79
Since Ore localization is an exact functor we get
TorCΓ
p (M ; DΓ) =
∼
=
TorpCG∗H (M ; (DG ∗ H)T −1 )
TorpCG∗H (M ; DG ∗ H) ⊗DG∗H DΓ
and conclude again with Lemma 9.4 that this module vanishes if p > 1.
(III)
The behaviour under directed unions remains to be checked. Let Γ =
S
Γ
Si∈I i be a directed union, then we know from Proposition 8.26 that DΓ =
i∈I DΓi , and Lemma 9.4 gives
∼
TorCΓ
p (M ; DΓ) =
=
=
colimi∈I TorpCΓi (resCΓ
CΓi M ; DΓ)
CΓ
i
colimi∈I TorCΓ
p (resCΓi M ; DΓi ) ⊗DΓi DΓ
0 for p > 1.
(ii) The proof is exactly the same, except that this time we start with the trivial
group and hence we can extend the vanishing results to p > 0, compare 7.3.
80
10
K-theory
In this section we collect information about the algebraic K-theory of our rings
CΓ, DΓ, N Γ and UΓ. Moreover, we compute K0 of the category of finitely
presented N Γ-torsion modules.
Every von Neumann algebra can be decomposed into a direct sum of algebras of
type I, II1 , II∞ and III, compare [82, Chapter V, Theorem 1.19]. Since a group
von Neumann algebra is finite only the types I and II1 can occur. So a priori
for a group Γ we have N Γ = N ΓI ⊕ N ΓII1 . Remember that a group is called
virtually abelian if it contains an abelian subgroup of finite index.
Proposition 10.1. Let Γ be a finitely generated group.
(i) If Γ is virtually abelian, then N Γ = N ΓI is of type I and can be written
as a finite direct sum
N Γ = ⊕kn=1 N ΓIn ,
where N ΓIn is an algebra of type In .
(ii) If Γ is not virtually abelian, then N Γ = N ΓII1 is of type II1 .
If Γ is not finitely generated mixed types can occur.
Proof. See page 122 in [36] and the references given there.
A von Neumann algebra of type In is isomorphic to Mn (L∞ (X; µ)) for some
measure space (X, µ). Note that K-theory is Morita invariant and compatible
with direct sums.
10.1
K0 and K1 of A and U
Let us start with K0 . As usual we fix a normalized trace trA on A. We denote
the center of A by Z(A). The center valued trace for A is a linear map
trZ(A) : A → Z(A),
which is uniquely determined by trZ(A) (ab) = trZ(A) (ba) for all a, b ∈ A and
trZ(A) (c) = c for all c ∈ Z(A), see [41, Proposition 8.2.8, p.517]. Our fixed
trace trA : A → R factorizes over the center valued trace, i.e. we have trA (a) =
trA (trZ(A) (a)) for all a ∈ A, compare [41, Proposition 8.3.10, p.525].
Proposition 10.2. Let A be a finite von Neumann algebra with center Z(A)
and let U be the associated algebra of affiliated operators. Let Z(A)sa denote
the vector space of selfadjoint elements in Z(A) and let Z(A)pos denote the
cone of nonnegative elements. Remember that Proj(A) denotes the monoid of
isomorphism classes of finitely generated projective A-modules.
(i) The natural map A → U induces an isomorphism
K0 (A)
∼
=K0 (U).
81
(ii) There is a commutative diagram
Proj(A)
- K0 (A)
@
@ dimA
@
@
R
@
?
- Z(A)sa
- R.
dimZ(A)
?
Z(A)pos
The map dimZ(A) is induced by the center valued trace. All maps in the
square are injective.
(iii) If A is of type II1 , then dimZ(A) is an isomorphism.
(iv) If A = Mn (L∞ (X; µ)), then the image of dimZ(A) consists of all measurable step functions on X with values in n1 Z, where we identify the center
of A with L∞ (X; µ).
Proof. (i) This has already been proven in Theorem 3.8.
(ii) In Section 3 we normalized the traces trMn (A) such that trMn (A) (1n ) = n.
We extend the center valued trace to matrices by setting
X
trn : Mn (A) → Z(A), (aij ) 7→
trZ(A) (aii ).
These maps are compatible with the stabilization maps Mn (A) → Mn+1 (A).
Since n1 trn has the characteristic properties of a center valued trace if we identify
Z(A) with Z(Mn (A)) we have
trn = n · trZ(Mn (A)) .
Two projections p and q are Murray von Neumann equivalent and therefore
isomorphic (compare Proposition 3.5) if and only if trZ(A) (p) = trZ(A) (q). The
center valued trace of a projection is nonnegative and selfadjoint, see [41, Theorem 8.4.3, p.532]. Therefore we get injective maps
LP roj (Mn (A))/∼
= → Z(A)pos
which are compatible with the stabilization maps. This gives an injective map
Proj(A) → Z(A)pos . Since the block sum of projections corresponds to an
ordinary sum of traces we see that it is a map of monoids and that Proj(A)
satisfies cancellation. Therefore the map to Proj(A) → K0 (A) is injective. This
implies that also the induced map dimZ(A) : K0 (A) → Z(A)+ is injective.
(iii) By [41, Theorem 8.4.4 (i), p.533] we know that the image of the center
valued trace in the case of a type II1 algebra is the set
{a ∈ Z(A) | a selfadjoint, nonnegative, |a| < 1}.
Because of the factor n in trn we get the result.
(iii) This follows from [41, Theorem 8.4.4 (ii), p.533].
82
In [56] the algebraic K1 -groups of von Neumann algebras are determined. Of
course they depend on the type of the von Neumann algebra. Moreover, the
authors compute a modified K1 -group K1w (A). Let us recall its definition.
Definition 10.3. Let R be a ring. Let K1inj (R) be the abelian group generated
by conjugation classes of injective endomorphisms of finitely generated free Rmodules modulo the following relations:
(i) [f ] = [g] + [h] if there is a short exact ladder of injective homomorphisms
- L
- M
- N
- 0
0
0
g
f
h
?
- L
?
- M
?
- N
- 0.
(ii) [f ◦ g] = [f ] + [g] if f and g are endomorphisms of the same module.
(iii) [idM ] = 0 for all modules M .
If R = A is a von Neumann algebra, then we define K1w (A) = K1inj (A).
Here w stands for weak isomorphism, compare Lemma 2.6. Note that if one
replaces the word injective endomorphism by isomorphism one gets a possible
definition of K1 (R). It turns out that the group K1w (A) admits a very natural
interpretation.
Proposition 10.4. There is a natural isomorphism
∼
=K1 (U).
K1w (A)
Proof. If we apply Lemma 2.6 to matrix algebras we see, that an endomorphism
f between finitely generated free A-modules is injective if and only if f ⊗ idU
is an isomorphism. Therefore f 7→ [f ⊗ idU ] gives a well-defined map since the
relations analogous to (i)–(iii) hold in K1 (U). To define an inverse we change
the point of view and consider K1 (U) as GL(U)ab . Let C be an invertible matrix
over U. Let T ⊂ A be the set of non-zerodivisors in A. From Proposition 13.7
we know that Mn (U) = Mn (A)(T · 1n )−1 and therefore we can find a matrix
A over A and s ∈ A such that C = As−1 1n . Note that A and s1n have to be
injective endomorphisms because they become invertible over U. We want to
show that the map
C 7→ [A] − [s1n ]
is well-defined. Suppose C = Bt−1 1n is another decomposition of the matrix.
The defining equivalence relation for the Ore localization tells us that there must
exist matrices u1n and v1n over R such that Au1n = Bv1n , s1n u1n = t1n v1n
and s1n u1n = t1n v1n ∈ T · 1n , compare 13.3. Since u and v have to be weak
isomorphisms one verifies, using relation (ii), that the map is well-defined. Since
K1w (A) is abelian the map factorizes over K1 (U). That the maps are mutually
inverse follows from the relations (i)–(iii).
83
An invertible operator a ∈ GLn (A) has a gap in the spectrum near zero, therefore we can define the Fuglede-Kadison determinant via the function calculus
as
1
detF K (a) = exp
trZ(A) (log(a∗ a)) .
2
The results from [56] can now be rephrased as follows.
Proposition 10.5. Let A be a finite von Neumann algebra and let U be the
associated algebra of affiliated operators.
(i) If A is of type II1 , then
K1 (U) = 0
and the Fuglede Kadison determinant gives an isomorphism
∼
=
detF K : K1 (A) - Z(A)×
pos ,
where Z(A)×
pos denotes the group of positive invertible elements in the center of A.
(ii) If A = Mn (L∞ (X; µ)) is of type In , then U = Mn (L(X; µ)) and we have
the following commutative diagram
- K1 (U)
K1 (A)
det
det
?
?
L∞ (X; µ)× - L(X; µ)× ,
where the vertical maps are isomorphisms.
Proof. See [56].
Let R be a ring. Sending an invertible element a ∈ R× to the class of the
corresponding 1 × 1-matrix in K1 (R) gives a map
(R× )ab → K1 (R).
Here (R× )ab is the abelianized group of units.
Proposition 10.6. Let A be a finite von Neumann algebra and let U be the
associated algebra of affiliated operators. The maps
(A× )ab → K1 (A)
and
(U × )ab → K1 (U)
are isomorphisms.
Proof. For the first map this is proven more generally for finite AW ∗ -algebras
in [35, Theorem 7]. Since we know that U is a unit-regular ring the statement
for the second map follows from [58] and [33].
84
10.2
Localization Sequences
Localizations yield exact sequences in K-theory. We will use such a sequence
to compute K0 of the category of finitely presented A-torsion modules. For
commutative rings or central localization the exact sequence already appears
in [2]. The case of an Ore localization is treated for example in [5]. Schofield
was able to generalize the localization sequence to universal localizations [78].
Since CΓ ⊂ DΓ is a universal localization for groups in C with a bound on the
order of finite subgroups we treat the localization sequence in this generality.
Let R be a ring and let Σ be a set of maps between finitely generated projective
R-modules. Let Σ be the saturation of Σ, compare page 101.
Definition 10.7. Given an injective universal localization R → RΣ we define
TR→RΣ to be the full subcategory of R-modules which are cokernels of maps
from Σ.
In the case of an Ore localization this category has a different description.
Note 10.8. If RT −1 is an Ore localization, then the category TR→RT −1 is
equivalent to the category of finitely presented torsion modules of projective
dimension 1.
Proof. Suppose M is a finitely presented torsion module of projective dimension
1. Then there is a short exact sequence 0 → P → Rn → M → 0 with P finitely
generated projective. Since − ⊗R RT −1 is exact and M ⊗R RT −1 = 0 the map
P → Rn is in Σ. On the other hand let f : P → Q be a map in Σ. Since
R → RΣ is injective f must be injective and therefore gives a 1-dimensional
resolution of coker(f ) by finitely generated projective R-modules.
Proposition 10.9 (Schofield’s Localization Sequence). Let R be a ring
and let Σ be a set of maps between finitely generated projective R-modules. Let
R → RΣ be a universal Σ-inverting ring homomorphism which is injective. Then
there is an exact sequence of algebraic K-groups
K1 (R) → K1 (RΣ ) → K0 (TR→RΣ ) → K0 (R) → K0 (RΣ ).
Proof. We will only explain the maps. For a proof see [78, Theorem 4.12]. The
simpler case of an Ore localization is also treated in [5]. Let a be an invertible
matrix over RΣ . The equality given in Proposition 13.11 induces the equality
[a] = [b] − [s]
in K1 (RΣ ) with s ∈ Σ̂ and b a matrix over R. Note that since a is invertible
also b is invertible over RΣ and therefore b ∈ Σ. Therefore it makes sense to
define
[a] 7→ [coker(s)] − [coker(b)] .
85
This gives the map K1 (RΣ ) → K0 (TR→RΣ ). Now suppose M = coker(f : P →
Q) with f ∈ Σ, then one defines
[M ] 7→ [Q] − [P ] .
This gives the map K0 (TR→RΣ ) → K0 (R). The other maps are the natural
maps induced by the ring homomorphism R → RΣ .
More information on localization sequences in algebraic K-theory can be found
in [29], [31], [9], [80], [86], [83] and [5].
If we apply this to a finite von Neumann algebra A we can compute K0 of the
category of finitely presented A-torsion modules, compare Subsection 3.3.
Proposition 10.10. Let A be a finite von Neumann algebra and let U be the
associated algebra of affiliated operators.
(i) The category TA→U is equivalent to the abelian category of finitely presented A-torsion modules.
(ii) If A is of type II1 , then K0 (TA→U ) = 0.
(iii) If A = Mn (L∞ (X; µ)) is of type In , then
K0 (TA→U ) = L(X; µ)/L∞ (X; µ).
Proof. (i) Over the semihereditary ring A every finitely presented module has
projective dimension 1.
(ii) This follows from the localization sequence since K0 (A) → K0 (U) is an
isomorphism by Theorem 3.8 and K0 (U) = 0 by Proposition 10.5(i).
(iii) Similarly with Proposition 10.5(ii).
Of course we can also apply the above sequence to the ring extension CΓ ⊂ DΓ
in the cases where we know that this is a universal localization.
Note 10.11. Let Γ be a group in the class C with a bound on the order of finite
subgroups. Then there is an exact sequence
K1 (CΓ) → K1 (DΓ) → K0 (TCΓ→DΓ ) → K0 (CΓ) → K0 (DΓ) → 0.
Proof. We have proven in Section 8 that for these groups the map
colimK∈F inΓ K0 (CK) → K0 (DΓ)
is surjective. This map factorizes over K0 (CΓ) and therefore the last map is
surjective.
Examples show that in general the map K0 (CΓ) → K0 (DΓ) is not even rationally injective. We also have an example of a group Γ which is not virtually
abelian with K1 (DΓ) 6= 0.
86
11
Appendix I: Affiliated Operators
The aim of this section is to prove that on the set U of closed densely defined
operators affiliated to the finite von Neumann algebra A there is a well-defined
addition and multiplication which turns U into an algebra. This fact goes back
to the first Rings of Operators paper of Murray and von Neumann from 1936
[63]. The proof given below essentially follows their proof.
For convenience we first collect a few facts about unbounded operators and fix
notation. Let H be a Hilbert space. An unbounded operator a : H ⊃ dom(a) →
H will simply be denoted by a. If b is an extension of a we write a ⊂ b,
i.e. dom(a) ⊂ dom(b) and the restriction of b to dom(a) coincides with a. Given
a, its graph is Γa = {(x, a(x))|x ∈ dom(a)} ⊂ H ⊕H. An operator is closed if its
graph is closed in H ⊕H. An operator is closable if the closure of its graph is the
graph of an operator. This operator is the minimal closed extension of a and will
be denoted by [a]. If a is densely defined a∗ can be defined as usual with domain
dom(a∗ ) = {x ∈ H| < a(.), x > is a continuous linear functional on dom(a)}.
If one adds or composes unbounded operators the domains of definition usually
shrink:
dom(a + b) = dom(a) ∩ dom(b)
dom(ab) = b−1 (dom(a)).
Here is a set of rules for unbounded operators.
Proposition 11.1. Let a,b and c be (unbounded) operators.
(i) (a + b)c = ac + bc.
(ii) a(b + c) ⊃ ab + ac, and equality holds if and only if a is bounded.
(iii) (ab)c = a(bc).
(iv) (a + b) + c = a + (b + c).
(v) If a ⊂ b, then ac ⊂ bc and ca ⊂ cb.
(vi) If a ⊂ b and c ⊂ d, then a + c ⊂ b + d.
(vii) If a is densely defined, then a∗ exists and is closed but not necessarily
densely defined.
(viii) If a is densely defined and closed, then a∗ exists and is densely defined
and closed.
(ix) Let a be densely defined. In order to show that a is closable it suffices
to prove a∗ to be densely defined, because then a∗∗ is the desired closed
extension.
(x) Let a be densely defined. Every extension b ⊃ a is densely defined and
b ∗ ⊂ a∗ .
87
(xi) Let a, b and ba be densely defined, then a∗ b∗ ⊂ (ba)∗ , and equality holds
if b is bounded.
(xii) Let a, b and a + b be densely defined, then a∗ + b∗ ⊂ (a + b)∗ .
A densely defined operator is called symmetric if s ⊂ s∗ , it is called selfadjoint
if s = s∗ . Note that by 11.1.(vii) a selfadjoint operator is automatically closed
(and densely defined by definition). If a is densely defined and closed, then a∗ a
is selfadjoint and the closure of the graph of a : dom(a∗ a) → H coincides with
the graph Γa of a. One says dom(a∗ a) is a core for a.
Lemma 11.2. If a is densely defined and closed, then the operator 1 + a∗ a has
a bounded inverse.
Proof. See [40, Theorem 2.7.8, p.158].
Given an operator a one defines the resolvent set as the set of those complex
numbers λ for which a bounded operator b (the resolvent) exists with b(λ − a) ⊂
(λ − a)b = idH . The complement of the resolvent set is the spectrum spec(a)
of the operator a. As for bounded selfadjoint operators there is a spectral
decomposition and a function calculus.
Proposition 11.3 (Spectral Decomposition). Let s be a selfadjoint operator (not necessarily bounded). Then spec(s) ⊂ R and there exists a unique
projection valued measure, i.e. a map es : Bspec(s) → B(H) from the Borel-σalgebra to B(H) with the following properties:
(i) For every Borel subset Ω ⊂ spec(s) the operator es (Ω) is a projection.
(ii) es (∅) = 0 and es (spec(s)) = idH .
(iii) For Borel sets Ω and Ω0 we have es (Ω ∩ Ω0 ) = es (Ω)es (Ω0 ).
(iv) es is strongly σ-additive, i.e. for
S a countable collection of pairwise disjoint
Borel sets (Ωn )n∈N with Ω = n∈N Ωn we have
lim es (
N →∞
N
[
Ωn )x = es (Ω)x
n=1
for all x ∈ H.
(v) A vector x ∈ H is in the domain dom(s) if and only if
Z
|λ|2 < es (dλ)x, x > < ∞,
R
where dλ denotes the Lebesgue measure on R.
(vi) For x ∈ dom(s) and arbitrary y ∈ H we have
Z
< s(x), y > =
λ < es (dλ)x, y > .
R
88
One often defines esλ = es ((−∞, λ]), and passing to a Stieltjes integral the last
fact is symbolically written as
Z ∞
s=
λdesλ
−∞
and called the spectral decomposition of s.
The representation of s as an integral can be used to define new operators which
are functions of s.
Proposition 11.4 (Function Calculus). Given a selfadjoint operator s and
a Borel measurable function f : spec(s) → R∪{±∞} such that es (f −1 ({±∞}) =
0, there exists a closed and densely defined operator f (s) with the following
properties.
(i) The vector x ∈ H is in the domain dom(f (s)) if and only if
Z
|f (λ)|2 < es (dλ)x, x > < ∞.
R
(ii) For every x ∈ dom(f (s)) and arbitrary y ∈ H we have
Z
< f (s)x, y > =
f (λ) < es (dλ)x, y > .
R
The assignment f 7→ f (s) is an essential homomorphism, i.e. it is a homomorphism of complex algebras if we define sum and product of (unbounded) operators
as the closure of the usual sum and product.
Proof. Compare [73, Section 5.3].
As an application one derives the polar decomposition for unbounded operators.
Proposition 11.5 (Polar Decomposition). Let a be densely defined and closed,
then there is a unique decomposition
a = us,
where u is a partial
isometry and s is a nonnegative selfadjoint operator. More
√
precisely s = a∗ a in the sense of function calculus, ker u = ker a, u∗ us = s,
u∗ a = s and uu∗ a = a.
Proof. [73, p.218]
Let now A ⊂ B(H) be a von Neumann algebra. We assume that 1A = idH .
Definition 11.6. If a is an operator and b ∈ B(H) is a bounded operator we
say that a and b commute if ba ⊂ ab. An operator a is affiliated to A if
it commutes with all bounded operators from the commutant A0 . A closed
subspace K ⊂ H is called affiliated to A if the corresponding projection pK
belongs to A. Given a set M of operators we define the commutant M 0 to be
the set of all bounded operators commuting with all operators in M .
89
Note that there is no condition concerning the domain of definition of an affiliated operator. Given a selfadjoint operator s it turns out that {s}0 is a
von Neumann algebra and that s is affiliated to the von Neumann algebra
W ∗ (s) = {s}00 . Similarly, if the closed and densely defined operator a is not
selfadjoint W ∗ (a) = ({a}0 ∩ {a∗ }0 )0 is a von Neumann algebra (see [73, p.206]).
Proposition 11.7. Let s be a selfadjoint operator. All spectral projections
es (Ω) lie in W ∗ (s) and all operators f (s) obtained via the function calculus
are affiliated to W ∗ (s).
Note that if s is affiliated to the von Neumann algebra A, then A0 ⊂ {s}0 and
therefore W ∗ (s) ⊂ A. The following lemma facilitates to verify whether a given
operator is affiliated.
Lemma 11.8. Let a be an operator and let M be a ∗-closed set of bounded
operators such that the algebra generated by M is dense in A0 with respect to
the weak topology. The following statements are equivalent.
(i) a is affiliated to A.
(ii) For all unitary operators u ∈ A0 we have ua ⊂ au.
(iii) For all unitary operators u ∈ A0 we have ua = au.
(iv) For all operators m ∈ M we have ma ⊂ am.
Proof. (iv) ⇒ (i): Since M is ∗-closed one can verify using 11.1(x) and 11.1(xi)
that also A∗ commutes with all elements of M . Hence M ⊂ {a}0 ∩ {a∗ }0 and
since {a}0 ∩ {a∗ }0 is a von Neumann algebra A0 ⊂ {a}0 ∩ {a∗ }0 .
(i) ⇒ (ii), (i) ⇒ (iv) and (iii) ⇒ (ii) are obvious.
Since every operator can be written as a linear combination of four unitary
operators (see [82, Proposition 4.9, p.20]) (ii) is a special case of (iv).
It remains to be shown that (ii) implies (iii). Suppose u is a unitary operator
and we have ua ⊂ au and u−1 a ⊂ au−1 . With 11.1(iii) and 11.1(v) we conclude
a ⊂ u−1 au and u−1 au ⊂ a and therefore a = u−1 au
In particular we see that in the case of a group von Neumann algebra N Γ an
operator on l2 Γ is affiliated to N Γ if it commutes with the right action of Γ
on l2 Γ. It is occasionally convenient to have the following reformulations of the
fact that an operator is affiliated.
Proposition 11.9. Let A ⊂ B(H) be a von Neumann algebra. Let a be a closed
densely defined operator on H. The following statements are equivalent.
(i) a is affiliated to A.
(ii) Let a = us be the polar decomposition, then u ∈ A and s ∈ U.
(iii) Let a = us be the polar decomposition, then u ∈ A and all spectral projections es (Ω) of the operator s are in A.
90
Proof. We only show that (i) implies (ii). Let a be affiliated to A. Let a = us
be the polar decomposition. If v is a unitary operator in A0 then vav −1 =
vuv −1 vsv −1 . Since vuv −1 is unitary and vsv −1 is selfadjoint it follows from the
uniqueness of the polar decomposition that vuv −1 = u and vsv −1 .
We now come back to our main object of study.
Definition 11.10 (The Algebra of Affiliated Operators). Let A be a finite von Neumann algebra. The set U is defined as the set of all operators a
with the following properties.
(i) a is densely defined.
(ii) a is closed.
(iii) a is affiliated to A.
As already mentioned, the aim of this section is to prove that there is a welldefined addition, multiplication and involution on U, such that U becomes a
complex ∗-algebra that contains A as a ∗-subalgebra. The only obvious algebraic
operation in U is the involution:
Proposition 11.11. If a ∈ U, then a∗ ∈ U.
Proof. According to 11.1.(viii) a∗ is closed, densely defined and for every b ∈ A0 ,
ba ⊂ ab implies b∗ a∗ ⊂ (ab)∗ ⊂ (ba)∗ ⊂ a∗ b∗ and therefore a∗ is affiliated to A
since A0 is ∗-closed.
To deal with the addition and composition one has to ensure that the domains
remain dense. At this point it is crucial that the von Neumann algebra A is
finite and one has a notion of dimension. So let us fix a trace tr. Remember
that a closed linear subspace P ⊂ H is called affiliated to A if the projection p
onto that subspace is affiliated to A. For closed affiliated subspaces there is a
dimension given by dim(P ) = tr(p).
Definition 11.12. Define for an arbitrary linear subspace L ⊂ H
dim(L) = sup{dim(P ) | P ⊂ H is closed and affiliated}
A linear subspace L ⊂ H is called essentially dense if dim(L) = 1.
This notion of dimension extends the usual notion of dimension if P ⊂ H is
closed and affiliated. One verifies that a subspace L is essentially dense if and
only if there is a sequence P1 ⊂ P2 ⊂ P3 ⊂ . . . ⊂ L of closed affiliated subspaces,
such that limi→∞ dim Pi = dim H = 1. This follows from the monotony of
the dimension for closed affiliated subspaces. The continuity property of the
dimension implies that an essentially dense subset is dense.
Lemma 11.13. The intersection of countably many essentially dense subspaces
is again essentially dense.
91
Proof. Let
T Lα , α = 1, 2, 3, . . . be a sequence of essentially dense subspaces and
let L = Lα . Let N > 0 be given. We will show that there exists a closed
affiliated subspace P ⊂ L with dim(P ) ≥ 1 − 1/2N . Choose PTα ⊂ Lα such
that dim Pα⊥ ≤ 1/2N +α. We will show by induction that dim( α≤n Pα )⊥ ≤
Pn
T
N +α
. There
into orthogonal subspaces α<n Pα =
Tα=1 1/2
T is a decomposition
⊥
α<n Pα ∩ Pn ⊕
α<n Pα ∩ (Pn ) which implies
dim
\
Pα
\
= dim
α<n
α≤n
≥ (1 −
= 1−
Pα − dim
n−1
X
α=1
n
X
\
α<n
Pα ∩ (Pn⊥ )
1/2N +α) − 1/2N +n
1/2N +α ,
α=1
and hence
dim(
\
α≤n
Pα )⊥ ≤
n
X
1/2N +α = 1/2N
n
X
α=1
α=1
1/2α ≤ 1/2N .
T
T
⊥
Note that
T α Pα ⊥is closed and affiliated. Since ( α Pα ) is the directed union
of the ( α≤n Pα ) we conclude from the continuity property of the dimension
T
T
T
⊥
≤ 1/2N . Since α Pα ⊂ α Lα and N was chosen arbitrarily
that dim( α Pα )T
we see that dim α Lα = 1.
Lemma 11.14. If a ∈ U and L ⊂ H is essentially dense, then a−1 (L) is
essentially dense. Moreover dom(a) is essentially dense for every a ∈ U.
Proof. Using the polar decomposition a = us one can treat the cases a = u
bounded and a = s selfadjoint separately. So let a be bounded. For every
closed affiliated subspace P ⊂ L, the subspace a−1 (P ) ⊂ a−1 (L) is closed since
a is continuous and affiliated. (It is sufficient to ensure that u(a−1 (P )) ⊂ a−1 (P )
for all unitary operators u from the commutant A0 .). Moreover, dim(a−1 (P )) ≥
dim(P ) and we see that dim(a−1 (L)) = 1.
R
Now let a = s be selfadjoint and nonnegative. Let s = λdeλ be the spectral
decomposition of s. Note that eλ H ⊂ dom(s) and seλ is a bounded operator.
The argument above shows that given 1 > 0 we can find a closed affiliated
subspace P ⊂ L with dim(seλ )−1 (P ) ≥ 1 − 1 . On the other hand given
any 2 > 0 we can find λ such that dim(eλ H)⊥ ≤ 2 . Now the orthogonal
decomposition (seλ )−1 (P ) = (seλ )−1 (P ) ∩ eλ H ⊕ (seλ )−1 (P ) ∩ (eλ H)⊥ implies
that dim(seλ )−1 (P ) ∩ eλ H ≥ 1 − (1 + 2 ). Since (seλ )−1 (P ) ∩ eλ H is closed,
affiliated and a subspace of s−1 (P ) we see that dim(s−1 (P )) = 1.
Let us now prove the second statement. If a = us is the polar decomposition,
then dom(a) = dom(s). Since eλ converges strongly to idH for λ → ∞ we see
that dim(dom(s)) = 1.
92
Lemma 11.15. If a ∈ U and b ∈ U, then a + b and ab are densely defined.
Proof. This follows immediately from the above lemmata.
Proposition 11.16. If a ∈ U and b ∈ U, then a + b and ab are affiliated and
closable. Moreover the closures [a + b] and [ab] are in U.
Proof. Using the list of rules 11.1 one checks that a + b is affiliated. Because of
11.11 we know that a∗ and b∗ are in U. The previous proposition shows that
a∗ + b∗ and therefore (a + b)∗ ⊃ a∗ + b∗ is densely defined. So a + b is closable
by 11.1.(ix). Remains to be verified that the closure [a + b] is again affiliated
to A. Now the condition ca ⊂ ac for a bounded operator c is equivalent to
(c ⊕ c)(Γa ) ⊂ Γa . Since c ⊕ c is continuous (c ⊕ c)(Γa ) ⊂ Γa and it follows that
the closure of an affiliated operator is again affiliated. The argument for ab is
similar.
So far we have a well-defined addition and multiplication. We still have to check
the axioms of an algebra with involution. The crucial point is that a closed
extension lying in U is unique. To prove this one first deals with symmetric
operators. Again it is important to have a well behaved notion of dimension.
Let us first collect a few facts about the Cayley transform of an operator.
Proposition 11.17 (Cayley Transform). Let s be a densely defined symmetric operator. Then the Cayley transform κ(s) is the operator (s − i)(s + i)−1
with domain dom(κ(s)) = (s + i)dom(s) and range im(κ(s)) = (s − i)dom(s).
This operator is an isometry (on its domain). Moreover κ(s)− 1 is injective and
im(κ(s) − 1) = dom(s). If in addition s is closed the subspaces (s ± i)dom(s)
are closed. The operator s is selfadjoint if and only if (s + i)dom(s) = H and
(s − i)dom(s) = H.
Proof. Compare [73, pages 203-205].
Proposition 11.18. Suppose that s is densely defined, closed, symmetric and
affiliated to A. Then s is already selfadjoint.
Proof. First note that all ingredients in the Cayley transform, namely κ(s),
dom(κ(s)) and im(κ(s)) are again affiliated to A. From the previous proposition
we know that im(κ(s) − 1) = dom(s). Let p = pdom(κ(s)) be the projection
onto dom(κ(s)) = (s + i)dom(s), then im(κ(s) − 1) = im(κ(s)p − p). From
(κ(s)p−p)∗ = p∗ (κ(s)p−p)∗ = p(κ(s)p−p)∗ we conclude that im((κ(s)p−p)∗ ) ⊂
im(p) and therefore dim((s + i)dom(s)) = dim(dom(κ(s))) = dim(im(p)) ≥
dim(im(κ(s)p − p)∗ ) = dim(im(κ(s)p − p)) = dim(dom(s)) = 1. Here we use the
fact that dom(s) is essentially dense. Since we also know that im((s + i)dom(s))
is a closed subspace it follows that this space is the whole Hilbert space. Since
κ(s) is an isometry and affiliated one can also verify that dim(im(κ(s))) = 1.
But im(κ(s)) is also closed and therefore it is the whole Hilbert space. This
means κ(s) is a unitary operator and therefore s is selfadjoint.
93
Proposition 11.19 (Unique Closure). Let a ⊂ b be two operators in U, then
a = b. In particular, if c is a closable operator whose closure [c] lies in U, then
[c] is the only closed extension of c lying in U.
Proof. Let b = us be the polar decomposition of b and set s0 = u∗ a ⊂ u∗ b = s.
Since u∗ is bounded s0 is still closed, densely defined and affiliated. It follows
that s0 ⊂ s = s∗ ⊂ s0∗ , i.e. s0 is symmetric. Now the previous proposition
implies s0 = s = s∗ = s0∗ and therefore dom(a) = dom(s0 ) = dom(s) = dom(b)
and a = b. The second assertion follows from the first. Note that [c] is the
minimal closed extension of c.
Theorem 11.20. Endowed with these structures U becomes a complex ∗-algebra
that contains A as a ∗-subalgebra.
Proof. One has to show that U with this addition multiplication and ∗-operation
fulfills the axioms of a ∗-algebra. As an example we will check the distributivity
law [a[bc]] = [[ab]+[ac]]. First one verifies similar to 11.15 and 11.16 that ab+ac
is densely defined and closable with closure afilliated to A. Now the rules for
unbounded operators imply ab + ac ⊂ [ab]+ [ac] and ab + ac ⊂ a(b + c) ⊂ a[b + c].
Taking the closure leads to [a[b + c]] ⊃ [ab + ac] ⊂ [[ab] + [ac]]. The above proven
uniqueness of a closure in U implies the desired equality.
94
12
Appendix II: von Neumann Regular Rings
In the first Rings of Operators paper from 1936 [63] Murray and von Neumann
studied the lattice of those subspaces of a Hilbert space which are affiliated
to a von Neumann algebra (ring of operators). Already in that paper they
introduced what we called the algebra of operators affiliated to a finite von
Neumann algebra. About the same time von Neumann axiomatized the system
of all linear subspaces of a given (projective) space in what he called a continuous
geometry [64]. The main new idea was that the notion of dimension (points,
lines, planes, ...) need not be postulated but could be proven from the axioms
and that dimensions need not be integer valued any more. Of course, the lattice
of affiliated subspaces was the main motivating example. He realized that the
algebra of affiliated operators gave a completely algebraic description of this
lattice in terms of ideals and idempotents. He singled out a class of rings which
should play the same role for an abstract continuous geometry and called these
rings regular [64] [65] [66].
There is an extensive literature on von Neumann regular rings, but for convenience we will collect some results in this section. More details and further
references can be found in [30].
Definition 12.1. A ring R satisfying one of the following equivalent conditions
is called von Neumann regular.
(i) For every x ∈ R there exists a y ∈ R, such that xyx = x.
(ii) Every principal right (left) ideal of R is generated by an idempotent.
(iii) Every finitely generated right (left) ideal of R is generated by an idempotent.
(iv) Every finitely generated submodule of a finitely generated projective module is a direct summand.
(v) Every right (left) R-module is flat.
(vi) TorR
p (M, R) = 0 for p ≥ 1 and an arbitrary right R-module M (R has
weak- (or Tor-) dimension zero.
(vii) Every finitely presented module is projective.
Proof. Compare [75, Lemma 4.15, Theorem 4.16, Theorem 9.15].
Several processes do not lead out of the class of von Neumann regular rings:
Proposition 12.2. (i) If R is von Neumann regular, then the matrix ring
Mn (R) is von Neumann regular.
(ii) The center of a von Neumann regular ring is von Neumann regular.
(iii) A directed union of von Neumann regular rings is von Neumann regular.
95
Proof. (iii) follows from 12.1(i). For (i) and (ii) see [30, Theorem1.7, Theorem
1.14].
In connection with localization the following properties of von Neumann regular
rings are of importance.
Proposition 12.3. Let R be a von Neumann regular ring.
(i) Every element a ∈ R is either invertible or a zerodivisor.
(ii) A von Neumann regular ring is division closed and rationally closed in
every overring.
(iii) If a von Neumann regular ring satisfies a chain condition, i.e. if it is
artinian or noetherian, then it is already semisimple.
Proof. (i) and (ii) are treated in 13.15. For (iii) see [30, p.21].
If our ring is a ∗-ring we can make the idempotent in Definition 12.1(ii) unique
if we require it to be a projection.
Definition 12.4. A von Neumann regular ∗-ring in which a∗ a = 0 implies
a = 0 is called ∗-regular.
The most important property of a ∗-regular ring is the following:
Proposition 12.5. In a ∗-regular ring
(i) Every principal right (left) ideal is generated by a unique projection.
(ii) Every finitely generated right (left) ideal is generated by a unique projection.
Proof. See [66, Part II, Chapter IV, Theorem 4.5].
Fortunately, in our situation we do not have to care to much about the additional
requirement in Definition 12.4.
Proposition 12.6. Let U be the algebra of operators affiliated to a finite von
Neumann algebra A. Any subring R of U which is ∗-closed and von Neumann
regular is already ∗-regular.
Proof. We only have to ensure that a∗ a = 0 implies a = 0. The proof is identical
to the one for 2.11.
Another refinement of the notion of von Neumann regularity that is used in
Section 3 is unit regularity. One should think of this condition as a finiteness
condition, compare [30, Section 4].
Definition 12.7. A ring R is unit regular, if for every x ∈ R there exists a unit
u ∈ R such that xux = x.
96
The following proposition due to Handelman characterizes unit regular rings
among all von Neumann regular rings.
Proposition 12.8. Let R be a von Neumann regular ring, then R is unit regular
if and only if the following holds: If L, P and Q are finitely generated projective
modules, then
P ⊕L∼
=Q⊕L
implies P ∼
= Q.
Proof. See [34].
In particular for such a ring the map from the semigroup of isomorphism classes
of finitely generated projective modules to K0 (R) is injective.
There seems to be no example of a ∗-regular ring which is not unit regular,
compare [30, p.349, Problem 48].
97
13
Appendix III: Localization of Non-commutative
Rings
In this section we will discuss different concepts of localizations for non-commutative
rings.
13.1
Ore Localization
Let R be a ring and let X ⊂ R be any subset. A ring homomorphism f : R → S
is called X-inverting if f (x) is invertible in S for every x ∈ X.
Definition 13.1. An X-inverting ring homomorphism R → RX is called universal X-inverting if it has the following universal property: Given any Xinverting ring homomorphism f : R → S there exists a unique ring homomorphism φ : RX → S such that the following diagram commutes.
RX
i
R
φ
f - ?
S.
As usual this implies that RX is unique up to a canonical isomorphism. One
checks existence by writing down a suitable ring using generators and relations.
Given a ring homomorphism R → S we denote by T(R → S) the set of all elements in R which become invertible in S. This set is multiplicatively closed. In
particular we have the multiplicatively closed set X = T(R → RX ). Obviously
X ⊂ X and one verifies that RX and RX are naturally isomorphic. Without
loss of generality we can therefore restrict ourselves to the case where X is multiplicatively closed, and in that case we usually write T instead of X. Moreover,
we will assume that T contains no zerodivisors since this will be sufficient for
our purposes and facilitates the discussion.
If the ring R is commutative it is well known that there is a model for RT
whose elements are fractions, or more precisely, equivalence classes of pairs
(a, t) ∈ R× T . If the ring is non-commutative the situation is more complicated.
The main difficulty is that once we have decided to work for example with right
fractions we have to make sense of a product like
at−1 a0 t0−1 .
Definition 13.2 (Ore Condition). Let T be a multiplicatively closed subset
of R. The pair (R, T ) satisfies the right Ore condition if given (a, s) ∈ R × T
there always exists (b, t) ∈ R × T such that at = sb.
A somewhat sloppy way to remember this is: For every wrong way (left) fraction
s−1 a there exists a right fraction bt−1 with s−1 a = bt−1 .
98
Proposition 13.3. Let R be a ring and let T ⊂ R be a multiplicatively closed
subset which contains no zerodivisors. Suppose the pair (R, T ) satisfies the right
Ore condition, then there exists a ring RT −1 and a universal T -inverting ring
homomorphism i : R → RT −1 such that every element of RT −1 can be written
as i(a)i(t)−1 with (a, t) ∈ R × T . The ring RT −1 is called a ring of right
fractions of R with respect to T .
Proof. Elements in RT −1 are equivalence classes of pairs (a, t) ∈ R×T . The pair
(a, t) is equivalent to (b, s) if there exist elements u, v ∈ R such that au = bv,
su = tv and su = tv ∈ S. For details of the construction and more information
see Chapter II in [81].
To avoid confusion we will usually call such a ring an Ore localization. There
is of course a left handed version of the above proposition. If both the ring of
right and the ring of left fractions with respect to T exist they are naturally
isomorphic since both fulfill the universal property. Since we assumed that T
contains no zerodivisors the map i is injective and we usually omit it in the
notation. We will frequently make use of the following fact.
Note 13.4. Finitely many fractions in RT −1 can be brought to a common
denominator.
Often one considers the case where T = NZD(R) is the set of all non-zerodivisors
in R. In that case, if the ring RT −1 exists, it is called the classical ring of
fractions (sometimes also the total ring of fractions) of R.
Example 13.5. Let Γ be the free group generated by {x, y}. The group ring
CΓ does not satisfy the Ore condition with respect to the set NZD(CΓ) of all
non-zerodivisors: Let Z ⊂ Γ be the subgroup generated by x. Now x − 1 is a
non-zerodivisor since it becomes invertible in UZ and therefore in the overring
UΓ of CΓ. (In fact every nontrivial element in CΓ is a non-zerodivisor since we
know from Section 6 that we can embed CΓ in a skew field.) The Ore condition
would imply the existence of (a, t) ∈ CΓ × NZD(CΓ) with (y − 1)t = (x − 1)a
alias
(x − 1)−1 (y − 1) = at−1 .
This implies that (−a, t)tr is in the kernel of the map (x − 1, y − 1) : CΓ2 → CΓ.
But this map is the nontrivial differential in the cellular chain complex C∗cell (EΓ)
where we realize BΓ as the wedge of two circles. Since EΓ is contractible the
map must be injective. A contradiction.
For us one of the most important properties of an Ore localization is the following.
Proposition 13.6. Let RT −1 be an Ore localization of the ring R, then the
functor − ⊗R RT −1 is exact, i.e. RT −1 is a flat R-module.
99
Proof. The functor is right exact. On the other hand there is a similar localization functor for modules which is naturally isomorphic to − ⊗R RT −1 and which
can be verified as being left exact. For more details see page 57 in [81].
We close this section with a statement about the behaviour of an Ore localization
under the passage to matrix rings.
Proposition 13.7. Suppose the pair (R, T ) satisfies the right Ore condition,
then also the pair (Mn (R), T · 1n ) fulfills the right Ore condition and there is a
natural isomorphism
Mn (R)(T · 1m )−1 ∼
= Mn (RT −1 ).
Proof. Let ((aij ), t · 1n ) ∈ Mn (R) × T · 1n be given. Since (R, T ) satisfies the
Ore condition and finitely many fractions in RT −1 can be brought to a common
denominator we can find ((bij ), s · 1n ) ∈ Mn (R) × T · 1n such that in Mn (RT −1 )
we have
(s−1 · 1n )(aij ) = (s−1 aij ) = (bij t−1 ) = (bij )t−1 · 1n .
We see that (Mn (R), T · 1n ) fulfills the Ore condition. Now since
Mn (R) → Mn (RT −1 )
is T · 1n -inverting the universal property gives us a map
Mn (R)(T · 1n )−1 → Mn (RT −1 ).
Since every matrix in Mn (RT −1 ) can be written as (aij )(t−1 · 1n ) with (aij ) ∈
Mn (R) the map is surjective and injective.
Corollary 13.8. The diagram
R · 1n
jRT −1 · 1n
i
?
Mn (R)
?
- Mn (RT −1 )
is a push-out in the category of rings and (unit preserving) ring homomorphisms.
Proof. Suppose we are given maps f : Mn (R) → S and g : RT −1 · 1n → S such
that the resulting diagram commutes, then since f ◦ i(T · 1n ) = g ◦ j(T · 1n ) we
see that f is T · 1n inverting. The universal property gives us a unique map
Mn (RT −1 ) ∼
= Mn (R)(T · 1n )−1 → S.
One verifies that the resulting diagram commutes.
100
13.2
Universal Localization
Instead of inverting a subset of the ring one can also invert a given set of matrices
or more generally some set of maps between R-modules. This shifts attention
from the ring itself to the additive category of its modules or some suitable
subcategory.
Let Σ be a set of homomorphisms between right R-modules. A ring homomorphism R → S is called Σ-inverting if for every map α ∈ Σ the induced map
α ⊗R idS is an isomorphism.
Definition 13.9. A Σ-inverting ring homomorphism i : R → RΣ is called
universal Σ-inverting if it has the following universal property: Given any Σinverting ring homomorphism R → S there exists a unique ring homomorphism
ψ : RΣ → S, such that the following diagram commutes.
RΣ
i
R
f
ψ
?
- S.
Note that given a ring homomorphism R → S the class Σ(R → S) of all those
maps between R-modules which become invertible over S need not be a set.
In the following we will therefore always restrict to the category FR of finitely
generated free R-modules or the category PR of finitely generated projective
R-modules. These categories have small skeleta, and we will always work with
those. In the first case we take the set of matrices as such a skeleton where the
object set is taken to be the natural numbers. In the second case we take as
objects pairs (n, P ) with n a natural number and P an idempotent n×n-matrix.
If we now look at the set Σ = Σ(R → RΣ ), the so-called saturation of Σ, we of
course have again that RΣ and RΣ are naturally isomorphic. The same holds
for every set Σ̂ with Σ ⊂ Σ̂ ⊂ Σ.
Proposition 13.10. The universal Σ-inverting ring homomorphism exists and
is unique up to canonical isomorphism.
Proof. Uniqueness is clear. In the case where Σ is a set of matrices existence
can be proven as follows. We start with the free R-ring generated by the set of
symbols {aij | (aij ) = A ∈ Σ} and divide out the ideal generated by the relations
given in matrix form by AA = AA = 1 where of course A = (aij ). If Σ is a set
of morphisms between finitely generated projectives see Section 4 in [78].
A set of matrices is lower multiplicatively closed if 1 ∈ Σ and a, b ∈ Σ implies
a 0
∈Σ
c b
101
for arbitrary matrices c of suitable size. We denote by Σ̂ the lower multiplicative
closure of a given set Σ. Note that Σ ⊂ Σ̂ ⊂ Σ and therefore RΣ ∼
= RΣ̂ . Two
matrices (or elements) a, b ∈ M(R) are called stably associated over R if there
exist invertible matrices c, d ∈ GL(R), such that
a 0
b 0
c
d−1 =
0 1n
0 1m
with suitable m and n. There is a similar definition for maps between finitely
generated projective modules. The following Proposition has a lot of interesting
consequences.
Proposition 13.11 (Cramer’s Rule). Let R be a ring and let Σ be a set of
matrices between finitely generated projective R-modules. Every matrix a over
RΣ satisfies an equation of the form
1 0
1 x
s
=b
0 a
0 1
with s ∈ Σ̂, x ∈ M (RΣ ) and b ∈ M (R). In particular every matrix over RΣ is
stably associated to a matrix over R.
Proof. See [78, Theorem 4.3].
Example 13.12. In the case of a commutative ring the universal localization
gives nothing new: Let R be a commutative ring and let Σ be a set of matrices
over R. Then RΣ = Rdet(Σ) , where det(Σ) is the set of determinants of the
matrices in Σ, because by Cramer’s rule, which is valid over any commutative
ring, a matrix is invertible if and only if its determinant is invertible.
Note 13.13. In general universal localization need not be an exact functor.
13.3
Division Closure and Rational Closure
Given a ring S and a subring R ⊂ S we denote by T(R ⊂ S) the set of all
elements in R which become invertible over S and by Σ(R ⊂ S) the set of all
matrices which become invertible over S. Both sets are multiplicatively closed.
One can consider abstractly the universal localizations RT(R⊂S) and RΣ(R⊂S) .
The question arises whether these rings can be embedded in S. Unfortunately
this is not always possible. The intermediate rings in the following definition
serve as potential candidates for embedded versions of RT(R⊂S) and RΣ(R⊂S) .
Definition 13.14. Let S be a ring.
(i) A subring R ⊂ S is called division closed in S if
T(R ⊂ S) = R× ,
i.e. for every element r ∈ R which is invertible in S the inverse r−1 already
lies in R.
102
(ii) A subring R ⊂ S is called rationally closed in S if
Σ(R ⊂ S) = GL(R),
i.e. for every matrix A over R which is invertible over S the entries of the
inverse matrix A−1 are all in R.
(iii) Given a subring R ⊂ S the division closure of R in S denoted by D(R ⊂ S)
is the smallest division closed subring containing R.
(iv) Given a subring R ⊂ S the rational closure of R in S denoted by R(R ⊂ S)
is the smallest rationally closed subring containing R.
Note that the intersection of division closed intermediate rings is again division
closed, and similarly for rationally closed rings. In [15, Chapter 7, Theorem 1.2]
it is shown that the set
{ai,j ∈ S | (ai,j ) invertible over S, (ai,j )−1 matrix over R }
is a subring of S and that it is rationally closed. Since this ring is contained in
R(R ⊂ S) the two rings coincide. From the definitions we see immediately that
the division closure is contained in the rational closure. The next proposition
is very useful if one has to decide whether a given ring is the division closure
respectively the rational closure.
Proposition 13.15. A von Neumann regular ring R is division closed and rationally closed in every overring.
Proof. Suppose a ∈ R is not invertible in R, then the corresponding right Rlinear map la : R → R is not an isomorphism. Therefore the kernel or the
cokernel is nontrivial. Both split off as direct summands because the ring is von
Neumann regular. The corresponding projection on the kernel or cokernel is
given by left-multiplication with a suitable idempotent. This idempotent shows
that a must be a zerodivisor. A zerodivisor cannot become invertible in any
overring. We see that R is division closed. A matrix ring over a von Neumann
regular ring is again von Neumann regular. Hence the same reasoning applied
to the matrix rings over R yields that R is rationally closed in every overring.
Since the division closure is contained in the rational closure the last statement
follows.
Note 13.16. In particular we see that once we know that the division closure
D(R ⊂ S) is von Neumann regular it coincides with the rational closure R(R ⊂
S).
The following proposition relates the division closure respectively the rational
closure to the universal localizations RT(R⊂S) and RΣ(R⊂S) .
Proposition 13.17. Let R ⊂ S be a ring extension.
103
(i) The map φ : RT(R⊂S) → S given by the universal property factorizes over
the division closure.
RT(R⊂S)
Φ
R
⊂
?
- D(R ⊂ S) ⊂- S
(ii) If the pair (R, T(R ⊂ S)) satisfies the right Ore condition, then Φ is an
isomorphism.
(iii) The map ψ : RΣ(R⊂S) → S given by the universal property factorizes over
the rational closure.
RΣ(R⊂S)
Ψ
R
⊂
?
- R(R ⊂ S)
⊂
- S
The map Ψ is always surjective.
Proof. (i) All elements of T(R ⊂ S) are invertible in D(R ⊂ S) by definition of
the division closure. Now apply the universal property. (ii) Note that T(R ⊂ S)
always consists of non-zerodivisors. Thus we can choose a ring of right fractions
as a model for RT(R⊂S) . Every element in imφ is of the form at−1 . Such an
element is invertible in S if and only if a ∈ T(R ⊂ S). We see that imφ is
division closed and Φ is surjective. On the other hand the abstract fraction
at−1 ∈ RT(R ⊂ S)−1 is zero if and only if a = 0 because T(R ⊂ S) contains no
zero divisors, so Φ is injective. (iii) That the map ψ factorizes over R(R ⊂ S)
follows again immediately from the universal property. By Cohn’s description
of the rational closure R(R ⊂ S) we need to find a preimage for ai,j where
(ai,j ) is a matrix invertible over S whose inverse lives over R. The generator
and relation construction of the universal localization given above immediately
yields such an element.
In general it is not true that the map Ψ is injective. In 13.21 we give a counterexample where S is even a skew field.
13.4
Universal R-Fields
We describe in this subsection a very special case of ring extensions, where the
rational closure R(R ⊂ S) is indeed isomorphic to the universal localization
RΣ(R⊂S) . Again the main ideas are due to Cohn [15].
Given a commutative ring R without zerodivisors there is always a unique field
of fractions. It is characterized by the fact that it is a field which contains R as a
104
subring and is generated as a field by R. One possibility for a non-commutative
analogue is the following definition. Note that following Cohn we use the terms
field, skew field and division ring interchangeably.
Definition 13.18. Let R be an arbitrary ring. A ring homomorphism R → K
with K a skew field is called an R-field. It is called an epic R-field if K is
generated as a skew field by the image of R. An epic R-field is called a ring of
fractions for R if the homomorphism is injective.
The problem with this definition is that a ring of fractions is not unique. In
13.21 we will see that the non-commutative polynomial ring C < x, u > admits
several non isomorphic rings of fractions. Let us again look at the commutative
case. The ring R = Z admits other epic R-fields than Q, namely the residue
fields Z → Fp . The diagram
RΣ(R→K)
R
?
- K
becomes in this special case
Z(p)
Z
?
- Fp ,
where Z(p) is the local ring ZT −1 with T = Z − (p). One interprets the map
Z(p) → Fp as a morphism from the Z-field Q to the Z-field Fp in a suitable
category, a so called specialization. There is a unique specialization from the Zfield Q to every other epic Z-field. The next definition takes these observations
as a starting point.
Definition 13.19. A local homomorphism from the R-field K to the Rfield L is an R-ring homomorphism f : K0 → L where K0 ⊂ K is an R-subring
of K and K0 − ker f = K0× . The ring K0 is a local ring with maximal ideal
ker f . Two local homomorphisms from K to L are equivalent if they restrict to
a common homomorphism which is again local. An equivalence class of local
homomorphisms is called a specialization. An initial object in the category of
epic R-fields and specializations is called a universal R-field. If moreover the
map R → K is injective it is called a universal field of fractions.
A universal R-field is unique up to isomorphism, but even if there exists a field of
fractions for R there need not exist a universal R-field ([15, p.395, Exercise 10]).
On the other hand, if a universal R-field exists the map need not be injective,
105
i.e. it need not be a universal field of fractions. Now given a ring of fractions
for R one can ask whether in the diagram
RΣ(R⊂K)
Ψ
R
ψ
?
- K
Ψ is an isomorphism. Note that if we have a ring extension R ⊂ K with K
a skew field, then D(R ⊂ K) is the subfield generated by R in K, and from
Proposition 13.15 we know that D(R ⊂ K) = R(R ⊂ K). In particular, since a
field of fractions is an epic R-field we have K = D(R ⊂ K) = R(R ⊂ K). As
we already know from the discussion in the last subsection Ψ is surjective. One
might hope that Ψ is an isomorphism:
(i) For every field of fractions R → K.
(ii) At least for universal fields of fractions R → K.
Unfortunately there are counterexamples in both cases. It is claimed in [4, p.332]
that there exists a counterexample to (ii). We give a concrete counterexample
to (i) in 13.21. After all these bad news about the non-commutative world now
a positive result due to Cohn. A ring is a semifir if it has invariant basis number
and every finitely generated submodule of a free module is free.
Proposition 13.20. Let R be a semifir, then there exists a universal field of
fractions and the map RΣ(R→K) → K is an isomorphism.
Proof. By definition of a semifir every finitely generated projective module is
free and has a well-defined rank. A map Rn → Rn between finitely generated
free modules is called full if it does not factorize over a module of smaller rank.
Now [15, Chapter 7, 5.11] says that there is a universal field of fractions R → K
such that every full matrix (alias map between finitely generated projectives)
becomes invertible over K. Since non full matrices can not become invertible
over K we see that the set of full matrices over R coincides with Σ(R → K).
Now [15, Chapter 7, Proposition 5.7 (ii)] yields that the map RΣ(R→K) → K is
injective.
The main examples of semifirs are non-commutative polynomial rings over a
commutative field and the group ring over a field of a finitely generated free
group [16, Section 10.9]. These rings are even firs. Before we go on, here the
promised counterexample. Compare [4].
Example 13.21. Let C < x, y, z > and C < x, u > be non-commutative
polynomial rings. The map defined by x 7→ x, y 7→ xu and z 7→ xu2 is injective. Since C < x, u > is a semifir there is a universal field of fractions
106
C < x, u >→ K. Now the division closure D of the image of C < x, y, z > in K
is a field of fractions for C < x, y, z >. Consider the diagram
C < x, y, z >Σ(C<x,y,z>⊂D)
Ψ
C < x, y, z >
?
- D
ψ
given by the universal property. The map Ψ is not injective since, for example,
the element y −1 z − x−1 y is mapped to zero. From Proposition 13.20 we know
that for the universal field of fractions of C < x, y, z > the corresponding map
is an isomorphism. So D is not the universal field of fractions for C < x, y, z >,
and in particular we have found two non-isomorphic fields of fractions for C <
x, y, z >.
13.5
Recognizing Universal Fields of Fractions
Let Γ be the free group on two generators. From the preceding subsection we
know that the group ring CΓ, which is a semifir, admits a universal field of fractions. The following gives a criterion to decide whether a given field of fractions
for CΓ, as for example D(CΓ ⊂ UΓ), is the universal field of fractions. First we
need some more notation. Let Γ be a group and let G be a finitely generated
subgroup. We say that G is indexed at t ∈ G if there exists a homomorphism
pt : G → Z which maps t to a generator. In that situation we have an exact
sequence
1 → ker pt → G → Z → 1
together with a homomorphic section given by 1 7→ t. Therefore the group ring
RG ⊂ RΓ has the structure of a skew polynomial ring.
RG = (R ker pt ) ∗ Z.
Definition 13.22. Let Γ be a free group. An embedding CΓ ⊂ D into a
skew field (e.g. a CΓ-field of fractions) is called Hughes-free if for every finitely
generated subgroup G and every t ∈ G at which G is indexed we have that the
set {ti | i ∈ Z} is D(C ker pt ⊂ D)-left linearly independent.
A particular case of the main theorem in [37] states:
Proposition 13.23. Any two CΓ-fields of fractions which are Hughes-free are
isomorphic as CΓ-fields.
This is useful since Lewin shows in [45] the following proposition.
Proposition 13.24. The universal field of fractions of the group ring CΓ of
the free group on two generators is Hughes-free.
107
Combining these results with Proposition 13.20 we get:
Corollary 13.25. Let Γ be a free group. If CΓ ⊂ D is a Hughes-free CΓ-field
of fractions, then it is a universal CΓ-field of fractions, and it is also a universal
localization with respect to Σ(CΓ ⊂ D).
13.6
Localization with Respect to Projective Rank Functions
So far we were mostly interested in universal localizations RΣ(R⊂S) where a ring
extension R ⊂ S was given. But there are other ways to describe reasonable
sets Σ one would like to invert. We are following Schofield [78].
Definition 13.26. Let R be a ring. A projective rank function for R is a
homomorphism ρ : K0 (R) → R such that for every finitely generated projective
R-module P we have ρ([P ]) ≥ 0 and ρ([R]) = 1. The rank function is called
faithful if ρ([P ]) = 0 implies P = 0.
Note that such a rank function is in general not monotone, i.e. P ⊂ Q does
not imply ρ(P ) ≤ ρ(Q) as it is the case for the dimensionfunction discussed in
Section 3. Given a projective rank function we define the inner rank of a map
α : P → Q between finitely generated projectives as
ρ(α) = inf{ρ(P 0 ) | α factorizes over the f.g. projective module P 0 }.
Definition 13.27. A map α : P → Q is called right full if ρ(α) = ρ(Q) and
left full if ρ(α) = ρ(P ). It is called full if it is left and right full.
Note 13.28. If R is semihereditary and the rank function is monotone, then
ρ(α) = ρ([imα]). If R is von Neumann regular, then every rank function is
monotone and could be called a dimension function if it is also faithful. In that
case the notions right full, left full and full correspond to surjective, injective
and bijective.
Again it is the free group which shows that in general these notions give something new.
Example 13.29. Let Γ be the free group on two generators x and y. Take ρ
as the composition
dim
ρ : K0 (CΓ) → K0 (UΓ) → R.
Now look at the map
CΓ2
(x−1,y−1)
−→
CΓ.
This is the differential in the cellular chain complex of the universal covering of
BΓ = S1 ∨ S1 . This map is right full, but not surjective, and it is injective, but
not left full with respect to ρ.
108
It is reasonable to invert universally all matrices (or maps between finitely generated projective modules) which are full with respect to a given projective rank
function ρ. The following lemma tells us that in one of our standard situations
we get nothing new.
Lemma 13.30. Let R → S be a ring homomorphism with S von Neumann
regular and suppose there is a faithful projective rank function dim : K0 (S) → R
for S. Let
ρ : K0 (R) → K0 (S) → R
be the induced projective rank function for R. Then a map α : P → Q which
becomes an isomorphism over S is a full map.
Proof. Suppose α ⊗R idS is an isomorphism. If α factorizes over the finitely
generated projective module P 0 , then in the induced factorization after tensoring
we have
ρ(α) ≤ ρ(P ) = dim(P ⊗R S) ≤ dim(P 0 ⊗R S).
Taking the infimum over all P 0 gives ρ(α) = ρ(P ) = ρ(Q). The map is full.
The following theorem of Schofield is very useful for our purposes:
Theorem 13.31. Let R be semihereditary and let ρ : K0 (R) → R be a faithful
projective rank function such that every map between finitely generated projectives factors as a right full followed by a left full map. If Σ is a collection of
maps that are full with respect to the rank function, then K0 (R) → K0 (RΣ ) is
surjective.
Proof. See [78, Theorem 5.2]. Note that over a semihereditary ring all rank
functions are automatically Sylvester projective rank functions by [78, Lemma
1.1, Theorem 1.11].
The interesting example of this situation is not the extension N Γ ⊂ UΓ, but
the extension CΓ ⊂ UΓ in the case where Γ is a free group or a finite extension
of a free group. Over a hereditary ring the condition about the factorization is
always fulfilled ([78, Corollary 1.17]). In general the following observation sheds
some light on this condition.
Note 13.32. Let R be semihereditary and ρ : K0 (R) → R be a projective
rank function with im(ρ) ⊂ 1l Z for some integer l. Every map between finitely
generated projectives factors as a right full followed by a left full map.
Proof. Let α : P → Q be a map between finitely generated projectives. Over a
semihereditary ring ρ(α) = inf{ρ(P 0 ) | imα ⊂ P 0 ⊂ P, P 0 f.g. projective} since
the map from an abstract P 0 to its image is always split. So there must be
a finitely generated projective submodule P 0 with im(α) ⊂ P 0 ⊂ Q such that
ρ(P 0 ) = ρ(α) = inf{ρ(P 0 )|im(α) ⊂ P 0 ⊂ Q}. In the factorization P → P 0 → Q
the first map is right full and the second is left full.
109
Let us summarize what we can achieve in this abstract set-up.
Corollary 13.33. Let R ⊂ S be a semihereditary subring of the von Neumann
regular ring S. Suppose there is a faithful dimension function K0 (S) → 1l Z for
some integer l. Then the diagram
K0 (RΣ(R⊂S) )
i∗
K0 (R)
Ψ∗
?
- K0 (R(R ⊂ S))
- K0 (S)
→
1
Z⊂R
l
is commutative and the map i∗ is surjective.
Unfortunately it is not clear whether or not Ψ∗ is surjective.
110
14
14.1
Appendix IV: Further Concepts from the Theory of Rings
Crossed Products
If H is a normal subgroup of the group Γ, the group ring CΓ can be described in
terms of the ring CH and the quotient G = Γ/H. This situation is axiomatized
in the notion of a crossed product. For a first understanding of the following
definition one should consider the case where the ring R is CG, the ring S is CΓ
and the map µ is a set theoretical section of the quotient map Γ → G = Γ/H
followed by the inclusion Γ ⊂ CΓ.
Definition 14.1 (Crossed Product). A crossed product R ∗ G = (S, µ) of
the ring R with the group G consists of a ring S which contains R as a subring
together with an injective map µ : G → S × such that the following holds.
(i) The ring S is a free R-module with basis µ(G).
(ii) For every g ∈ G conjugation map
cµ(g) : S → S,
s 7→ µ(g)sµ(g)−1
can be restricted to R.
(iii) For all g, g 0 ∈ G the element
τ (g, g 0 ) = µ(g)µ(g 0 )µ(gg 0 )−1
lies in R× .
In this definition we consider a crossed product as an additional structure on
the given ring S = R ∗ G. Alternatively one can consider a crossed product as a
new ring constructed out of a given ring R, a group G and some twisting data.
This will be the content of Proposition 14.2 below.
The first condition in the above definition tells us that every element of R ∗ G
can be written in the form
X
rg µ(g),
with rg ∈ R,
g∈G
where only finitely many of the rg are nonzero. The multiplication is determined
by
rµ(g) · r0 µ(g 0 ) = rcµ(g) (r0 )τ (g, g 0 )µ(gg 0 ).
By associativity [rµ(g) · r0 µ(g 0 )] · r00 µ(g 00 ) = rµ(g) · [r0 µ(g 0 ) · r00 µ(g 00 )]. Writing
this out yields the following equation in R
rcµ(g) (r0 ) (cτ (g,g0 ) ◦ cµ(gg0 ) )(r00 ) τ (g, g 0 )τ (gg 0 , g 00 ) =
rcµ(g) (r0 ) (cµ(g) ◦ cµ(g0 ) )(r00 ) cµ(g) (τ (g 0 , g 00 ))τ (g, g 0 g 00 ).
111
Setting r = r0 = r00 = 1 yields
τ (g, g 0 ) · τ (gg 0 , g 00 ) = cµ(g) (τ (g 0 , g 00 )) · τ (g, g 0 g 00 ).
Setting r = r0 = 1 and cancelling the τ ’s yields
cτ (g,g0 ) ◦ cµ(gg0 ) = cµ(g) ◦ cµ(g0 ) .
On the other hand these two equations imply the above. This leads to the
following proposition.
Proposition 14.2 (Construction of a Crossed Product). Given a ring R,
a group G and maps
c : G → Aut(R),
τ : G × G → R× ,
g 7→ cµ(g) ,
such that
τ (g, g 0 ) · τ (gg 0 , g 00 )
cτ (g,g0 ) ◦ cµ(gg0 )
= cµ(g) (τ (g 0 , g 00 )) · τ (g, g 0 g 00 )
= cµ(g) ◦ cµ(g0 )
holds, we can construct a ring R∗τ ,c G together with a map µ : G → R∗τ ,c G such
that µ is a crossed product structure on R∗τ ,c G and cµ(g) = cµ(g) and τ = τ .
Proof. Take the free R-left module on symbols µ(g) and define a multiplication
as above. The conditions make sure that this multiplication is associative.
Usually one omits τ , c and µ in the notation. According to the situation we
will write R ∗ G, (R ∗ G, µ) or R∗τ,c G. The following proposition describes
homomorphisms out of a crossed product.
Proposition 14.3. Let R∗τ,c G be a crossed product and let S be a ring. Suppose we are given a ring homomorphism f : R → S and a map ν : G → S ×
such that
G×G
τ
- R
@
@
@
τν @
@ ?
R
S
R
and
- S
cµ(g)
?
R
cν(g)
?
- S
commute. Then there exists a unique ring homomorphism
F : R∗τ,c G → S
such that F extends f and F ◦ µ = ν. Conversely a homomorphism R∗τ,c G → S
determines f and ν (and therefore τν and cν ).
112
In particular if we apply this to the case where S is a crossed product we get
the notion of a crossed product homomorphism. Again we can either consider
such a crossed product homomorphism f ∗ G : R ∗ G → R̃ ∗ G as a usual ring
homomorphism with additional properties or as a new ring homomorphism built
out of f : R → R̃.
Proposition 14.4 (Crossed Product Homomorphism). Let (R∗G, µ) and
(R̃ ∗ G, µ̃) be crossed products. A ring homomorphism F : R ∗ G → R̃ ∗ G is
called a crossed product homomorphism if
(i) The following diagram commutes
R∗G
µ
G
F
?
µ̃R̃ ∗ G.
(ii) The map F restricts to a map R → R̃.
On the other hand, given a ring homomorphism f : R → R̃ such that
G×G
τ
- R
@
@
f
@
τ̃ @
R
@ ?
R̃
R
and
cµ(g)
?
R
- R̃
c̃ν̃(g)
?
- R̃
commute, there exists a unique crossed product homomorphism
f ∗ G : R∗τ,c G → R̃∗τ̃ ,c̃ G
which extends f .
The following propositions in the main text also deal with crossed products:
Proposition 8.5 on page 56 (Crossed products and localizations), Proposition 8.12
on page 62 (Crossed products and chain conditions) and Lemma 9.4 on page
78.
14.2
Miscellaneous
In this subsection we collect some definitions from ring theory which are used
throughout the text.
Definition 14.5. Let R be a ring and f : Rn → Rm be any R-linear map
between finitely generated free right R-modules. The following can be taken
as a definition of the notions coherent, semihereditary, semifir, von Neumann
regular.
113
(i) R is coherent, if im(f ) is always finitely presented.
(ii) R is semihereditary, if im(f ) is always projective.
(iii) R is a semifir, if im(f ) is always a free module and the ring has invariant
basis number.
(iv) R is von Neumann regular, if im(f ) is always a direct summand of Rm .
A ring R has invariant basis number if every finitely generated R-module
has a well-defined rank, i.e. Rn ∼
= Rm implies n = m.
Definition 14.6. (i) A ring R is called prime if there are no zerodivisors on
the level of ideals, i.e. if I and J are two sided ideals then IJ = 0 implies
I = 0 or J = 0.
(ii) A ring R is called semiprime if there are no nilpotents on the level of
ideals, i.e. if I is a two sided (or one sided) ideal, then I n = 0 implies
I = 0. It is sufficient to check the case n = 2.
A semiprime ring is a subdirect product of prime rings [76, p.164]. The remarks
on page 186 in [76] might help to relate semiprimeness to other ring theoretical
concepts. For the notions of artinian, noetherian and semisimple rings we
recommend pages 495–497 in [76] where one finds a list with characterizations
and basic properties of these notions.
114
Glossary of Notations
(A) and (B)
(C)
see 5.10 and 8.3
see 8.4
Γ
G
H
g,h or k
e
G∗H
Aut(G)
1
|F inΓ| Z
X or Y
EG
’generic’ group
’generic’ subgroup of Γ
’generic’ quotient of Γ
’generic’ element of a group
unit of a group
free product of groups
automorphism group of G
see page 26
’generic’ class of groups
class of elementary amenable groups,
see Definition 7.1
Linnell’s class of groups, see Definition 7.5
groups in C with a bound on the order
of finite subgroups
class of groups not containing Z ∗ Z
Groups
C
C0
NF
Rings
R or S
R⊂S
R×
Mn (R)
GLn (R)
e or f
p or q
T
NZD(R)
T(R → S)
T(R ⊂ S)
Σ
Σ(R → S)
Σ(R ⊂ S)
RX
’generic’ ring
’generic’ ring extension
multiplicative group of units
ring of n × n matrices over R
group of invertible n × n-matrices over
R
’generic’ idempotent
’generic’ projection in a ∗-ring
’generic’ multiplicatively closed subset
of R
set of all non-zerodivisors in R
see remarks after Definition 13.1
special case with R → S embedding of
a subring
’generic’ set of matrices over R
see remarks after Definition 13.9
special case with R → S embedding of
a subring
universal X-inverting ring, see Definition 13.1
115
RT
RΣ
D(R ⊂ S)
R(R ⊂ S)
R∗G
µ
cr
τ (g, g 0 )
f ∗G
see remarks after Definition 13.1
universal Σ-inverting ring, see Definition 13.9
division closure of R in S, see Definition 13.14
ratioonal closure of R in S, see Definition 13.14
crossed product of R with G, see Definition 14.1
crossed product structure map, see Definition 14.1
conjugation map x 7→ rxr−1
see Definition 14.1
crossed product homomorphism, see
14.4
Specific rings
CΓ
RΓ
A
NΓ
U
UΓ
SΓ
T(Γ)
Σ(Γ)
D(Γ)
R(Γ)
B(H)
M0
M 00
W ∗ (s)
L∞ (X, µ)
L(X, µ) = L(X)
group ring with complex coefficients
group ring with R-coefficients, see Conjecture 5.3
’generic’ von Neumann algebra, see
page 5
group von Neumann algebra, see page 2
algebra of operators affiliated to a A,
see Definition 11.10
algebra of operators affiliated to N Γ,
see 2.1
see Theorem 5.10
see page 52
see page 52
see page 52
see page 52
algebra of bounded operators
commutant, see page 5
double commutant
see page 90
algebra of essentially bounded functions
algebra of measurable functions, see
Example 2.3
Modules
M
P or Q
tM
TM
PM
’generic’ module
’generic’ finitely generated module
torsion submodule, see Definition 3.14
largest zero-dimensional submodule,
see Definition 3.15
see Definition 3.15
116
Homology
HpΓ (X; N Γ)
HpΓ (X; UΓ)
(2)
bp (Γ)
χ(Γ)
χvirt (Γ)
χ(2) (Γ)
TorR
p (−; M )
see page 24
see Definition 4.1
L2 -Betti numbers, see page 24
Euler characteristic of BΓ
virtual Euler characteristic, see page 28
L2 -Euler characteristic, see page 29
derived functors of − ⊗R M
K0 (R)
Grothendieck group of isomorphism
classes of f.g. proj. R-modules
see page 69
see Definition 10.3
see Definition 10.3
see Definition 10.3
see 5.3
K-Theory
G0 (R)
K1 (R)
K1inj (R)
K1w (A)
colimK∈F inΓ K0 (CK)
Lattices
L
LP roj (A)
’generic’ lattice, see page 10
lattice of projections in a von Neumann
algebra
lattice of closed Hilbert A-submodules
of M
lattice of all submodules of M
set of all finitely generated submodules
of PR
set of all submodules of PR which are
direct summands
LHilb (M )
Lall (MR )
Lf g (PR )
Lds (PR )
Miscellaneous
E(Γ, F in)
HC0 (R)
H
< x, y >
l2 X
l2 (A)
L2 (X; µ)
see page 33
see proof of Proposition 5.19
’generic’ Hilbert space
scalar product on a Hilbert space
Hilbert space with orthonormal basis X
see page 10
Hilbert space of square integrable functions
ideal of finite rank operators in B(H)
ideal of trace class operators
Schatten ideals, see page 37
L0 (H)
L1 (H)
Lp (H)
117
tr(a)
ρ, ρ±
Ap
N Γ0
N Γp
trA
trZ(A)
trMn (A)
dimU
dim0U
detF K
TR→RΣ
p ∼MvN q
dom(a)
a⊂b
spec(a)
κ(s)
Proj(R)
trace of a trace class operator
see Definition 6.1
see Proposition 6.3
see Proposition 6.3
see Proposition 6.3
trace of a finite von Neumann algebra,
see page 5
center valued trace, see page 81
see page 15
see Proposition 3.9 and Theorem 3.12
see Definition 3.10
Fuglede-Kadison determinant,
see
page 84
see Definition 10.7
Murray von Neumann equivalence of
projections, see Lemma 3.6
domain of an operator
b is an extension of a, see page 87
spectrum of an operator, see page 88
Cayley transform of s, see Proposition 11.17
monoid of isomorphism classes of
f.g. proj. R-modules
118
Index
AW ∗ -algebra, 14
K1 of, 84
G-theory, 69
G0
and K0 , 69
of an Ore localization, 69
K-theory, 81
and localization, 85
negative, 34
K0
and G0 , 69
of algebra of affiliated operators,
81
of finitely presented torsion modules, 86
of unit regular ring, 97
of von Neumann algebra, 81
K1
of AW ∗ -algebra, 84
of algebra of affiliated operators,
84
of injective endomorphisms, 83
of unit regular ring, 84
of von Neumann algebra, 84
L2 -Euler characteristic, 29
L2 -homology, 24
universal coefficient theorem for,
77
R-field, 105
epic, 105
X-inverting ring homomorphism, 98
Γ-space, 24
with finite isotropy, 77
Σ-inverting ring homomorphism, 101
∗-closed
subrings of UΓ, 62
∗-regular ring, 9
definition of, 96
affiliated operator
characterization of, 90
algebra
AW ∗ , 14
matrix, 14
von Neumann, 5
algebra of affiliated operators
K0 of, 81
K1 of, 84
definition of, 91
almost equivariant, 40
amenable
group, 44
von Neumann algebra, 45
amenable group
characterization of, 45
Folner criterion, 45
arbitrary module, 21
dimension for, 17
artinian ring, 62, 114
and crossed product, 62
assembly map, 33
Atiyah conjecture, 26
and directed unions, 27
and division closure, 32
and subgroups, 28
for the free group on two generators, 42
for torsionfree groups, 32
iff DΓ skew field, 74
strategy for, 29
with coefficients, 27
Atiyah-Hirzebruch spectral sequence,
33
Bass trace map, 35
Bredon homology, 33
category
of finitely generated free modules, 101
of finitely generated projective
modules, 101
abelianized group of units, 84
affiliated
operator, 89
subspace, 89
119
of finitely presented torsion modules, 85
orbit, 33
Cayley
graph, 39, 45
transform, 93
center
of von Neumann regular ring,
95
valued dimension, 82
valued trace, 81
chain condition
and von Neumann regular rings,
96
Chern character, 35
classes of groups, 43
classical ring of fractions
definition of, 99
of CΓ, 53
classifying space
for a family, 33
closed
extension of an operator, 87
operator, 87
closure
division, 32, 52, 103
rational, 32, 52, 103
cofinal-measurable, 22
coherent ring, 11, 20
definition of, 114
common complement, 13
commutant, 5, 89
commutative ring
field of fractions of, 104
commute, 89
complement
common, 13
complete lattice, 10, 11, 16
composition
of unbounded operators, 87
condition
Ore, 8
conjecture
Atiyah, 26
Atiyah with coefficients, 27
isomorphism, 33
Kadison, 37
Serre’s, 28
zero divisor, 28
construction
GNS, 10
core of an operator, 88
counterexample
AG 6= N F , 45
EG =
6 AG, 45
SΓ not semisimple, 31
SΓ not unique, 32
field of fractions not universal,
106
injective but not left full, 108
non-isomorphic fields of fractions,
106
nontrivial module with vanishing dimension, 22
right full but not surjective, 108
to Ψ injective, 106
to Ψ injective for universal field
of fractions, 106
to Ore condition, 99
Cramer’s rule, 60, 85, 102
crossed product
and artinian rings, 62
and localization, 56
and noetherian rings, 62
and polycyclic-by-finite groups,
62
and semiprime rings, 62
construction of, 112
definition of, 111
homomorphism, 113
semisimple, 63
dense
essentially, 91
determinant
Fuglede-Kadison, 84
dimension, 15
additivity of, 17
center valued, 82
cofinality of, 17
of affiliated subspace, 91
of arbitrary U-module, 17
120
for EG =
6 AG, 45
group in C, 46
infinite cyclic group, 3
injective but not left full, 108
non-isomorphic fields of fractions,
106
nontrivial module with vanishing dimension, 22
not satisfying Ore condition, 99
of semifirs, 106
right full but not surjective, 108
universal localization of commutative ring, 102
extension
and localization, 58
of an operator, 87
of finitely generated projective
A-module, 15
of finitely generated projective
U-module, 15
of Hilbert A-module, 15
projective, 85
Tor-, 6
weak-, 6
directed union, 4, 71
and Atiyah conjecture, 27
division closed
definition of, 102
division closure, 32, 52
and Atiyah conjecture, 32
definition of, 103
division ring, 105
domain
of an operator, 87
double commutant theorem, 5
faithful trace, 5, 15
field, 105
field of fractions
of a commutative ring, 104
universal, 105
finite homological type, 28
finite trace, 5
finitely generated projective module,
20
finitely presented module, 20
finitely presented torsion modules,
85
K0 of, 86
flat, 8, 30, 45, 99
fraction
of operators, 7
Fredholm module, 37, 38
p-summable, 38
construction of, 40
free
Hughes-, 107
free group, 74
and DΓ, 74
finite extensions of, 75
free group on two generators, 37, 99
Atiyah conjecture for, 42
geometric properties of, 39
not amenable, 44
free product, 49
elementary amenable groups, 43
and TorCΓ
p (−; DΓ), 77
and Ore localization, 53
elements
stably associated, 60
epic R-field, 105
equivalent
Murray von Neumann, 13
equivariant
almost, 40
essential
homomorphism, 89
essentially bounded function, 3, 6
essentially dense, 91
Euler characteristic, 28
L2 -, 29
for groups in C, 78
virtual, 28
exact functor, 8, 99
example
Ψ not injective, 106
SΓ not semisimple, 31
SΓ not unique, 32
field of fraction not universal,
106
for AG 6= N F , 45
121
L2 -, 24
Bredon, 33
generalized equivariant, 33
reduced L2 -, 24
unreduced L2 -, 24
with twisted coefficients, 24
homomorphism
crossed product, 113
essential, 89
local, 105
Hughes-free, 107
Fuglede-Kadison determinant, 84
full, 108
function
essentially bounded, 3, 6
measurable, 6
square integrable, 6
step, 82
function calculus, 89
functor
exact, 8, 99
left exact, 20
Fölner criterion, 45
idempotent, 4, 12, 28
induction
set-up, 61
transfinite, 43, 46
induction map, 29
induction principle
for C, 47
for EG, 44
induction theorem
Moody’s, 36, 70
infinite cyclic group, 3
infinite locally finite group, 31
invariant basis number, 114
invariant mean, 44, 45
invertible operator, 7
isomorphism
lattice, 10, 11
weak, 7, 83
isomorphism classes of projections,
15
isomorphism conjecture, 33, 76
consequences of, 34
isomorphism of projections, 13
iterated localization, 59
GNS construction, 10
Goldie’s theorem, 64
graph
Cayley, 39
of an operator, 87
group
amenable, 44
elementary amenable, 43
free, 74
free on two generators, 37
infinite cyclic, 3
infinite locally finite, 31
of units, 84
torsionfree, 26, 31
virtually abelian, 81
virtually cyclic, 36
group von Neumann algebra, 1
of infinite cyclic group, 3
operators affiliated to, 90
type of, 81
groups
X -by-Y, 43, 53
classes of, 43
locally-X -, 43, 53
K-theory, 14
Kadison conjecture, 37
Künneth spectral sequence, 77
Hattori-Stallings rank, 35
hereditary ring, 76, 109
Hilbert A-module, 10
Hilbert A-modules
lattice of, 11
homological type
finite, 28
homology
lattice, 10
complete, 10, 11, 16
of projections, 11
lattice isomorphism, 10, 11
lattice of Hilbert A-modules, 11
122
Laurent polynomial ring, 3
skew-, 63, 64
left exact functor, 20
left full
definition of, 108
left regular representation, 10
Linnell
’s class of groups, 45
Linnell’s class of group, 45
Linnell’s class of groups
and DΓ, 52
and RΓ, 52
and TorCΓ
p (−; DΓ), 77
and Euler characteristics, 78
and free products, 46
examples, 46
Linnell’s theorem, 52
local homomorphism, 105
localization
and K-theory, 85
and crossed products, 56
and extensions, 58
iterated, 59
Ore, 8, 64
sequence, 85
lower multiplicatively closed, 101
torsionfree, 19
modules
finitely generated free, 101
finitely generated projective, 101
Moody’s induction theorem, 36, 70
Murray von Neumann equivalent, 13
negative K-theory, 34
noetherian ring, 64, 114
and crossed products, 62
non-zerodivisor, 4, 7
in UG ∗ Z, 64
non-zerodivisors
of CΓ, 52
normal trace, 5, 15
normalization of the trace, 15
Novikov-Shubin invariants, 25
operator
affiliated, 89
closed, 87
core of, 88
domain of, 87
finite rank, 37
graph of, 87
invertible, 7, 84
selfadjoint, 88
spectrum of, 88
symmetric, 88
trace class, 37
unbounded, 87
unique closure of, 94
operators
affiliated to a group von Neumann algebra, 90
orbit category, 33
tensor product over, 34
Ore
condition, 8
localization, 8
Ore condition
definition of, 98
for CΓ, 53
Ore localization
and G-theory, 69
and matrix rings, 100
map
stabilization, 14
matrices
stably associated, 60
matrix algebra, 14
matrix ring
and Ore localization, 100
of a von Neumann regular ring,
95
mean
invariant, 44
measurable function, 6
module, 4
arbitrary, 21
finitely generated projective, 20
finitely presented, 20
Fredholm, 37, 38
Hilbert A-, 10
torsion, 19, 21, 22, 85
123
and noetherian semiprime rings,
64
definition of, 99
is exact, 99
of CΓ, 53
left regular, 10
very trivial, 40
resolvent set, 88
right full
definition, 108
ring, 4
∗-regular, 9
artinian, 62, 114
coherent, 11, 20, 114
division, 105
hereditary, 76, 109
homomorphism, 4
noetherian, 64, 114
prime, 114
semihereditary, 11, 20, 30, 114
semiprime, 62, 64, 114
semisimple, 31, 62, 114
unit regular, 8, 13
von Neumann regular, 6, 11, 29,
95, 114
ring homomorphism
X-inverting, 98
Σ-inverting, 101
universal Σ-inverting, 101
universal X-inverting, 98
ring of fractions, 99
classical, 99
definition of, 105
rule
Cramer’s, 60, 85, 102
polar decomposition, 89
polycyclic-by-finite group
and crossed product, 62
polynomial
Laurent, 3
prime ring
definition of, 114
processes
C closed under, 46
product
free, 49
of unbounded operators, 6, 87
projection, 4, 12, 13
valued measure, 88
projections
isomorphism classes of, 15
isomorphism of, 13
lattice of, 11
Murray von Neumann equivalence of, 13
projective dimension, 85
projective rank function, 76
definition of, 108
push-out
of rings, 66, 100
saturation, 101
Schatten ideals, 37
selfadjoint operator, 88
semifir, 75, 106
and universal field of fractions,
75
definition of, 114
examples of, 106
universal field of fractions of,
106
semihereditary ring, 11, 20, 30
and finitely presented modules,
86
definition of, 114
semiprime
rank
Hattori-Stallings, 35
rank function
faithful, 108
monotone, 108
over semihereditary ring, 109
projective, 76, 108
Sylvester projective, 76
rational closure, 32, 52
definition of, 103
rationally closed
definition of, 103
reduced L2 -homology, 24
representation
124
subring of UΓ, 62
semiprime ring, 62, 64
definition of, 114
semiprime rings
and crossed products, 62
semisimple ring, 31, 62, 114
structure theory of, 64
Serre’s conjecture, 28
skew field, 105
skew-Laurent polynomial ring, 63,
64
specialization, 105
spectral sequence
Atiyah-Hirzebruch, 33
Künneth, 77
spectrum
gap in, 84
of an operator, 88
square integrable function, 6
stabilization map, 14
stably associated, 60, 102
step function, 82
Stieltjes integral, 89
strong topology, 5
subrings
∗-closed, 62
subspace
affiliated, 89
essentially dense, 91
sum
of unbounded operators, 6, 87
summability
of Fredholm module, 38
symmetric operator, 88
torsion element, 20
torsion module, 21, 22
torsionfree group, 26, 31
and Atiyah conjecture, 32
torsionfree module, 19
trace, 5, 91
center valued, 81, 82
faithful, 5, 15
finite, 5
normal, 5, 15
normalization of, 15
of a group von Neumann algebra, 5
transfinite induction, 43, 46
tree, 39
ultraweak topology, 5
unbounded operator
commuting with bounded, 89
definition of, 87
domain of, 87
extension of, 87
graph of, 87
spectrum of, 88
unbounded operators
composition of, 87
product of, 87
sum of, 87
union
directed, 4, 71
unit regular ring, 8, 13
K0 of, 97
K1 of, 84
definition of, 96
units
abelianized group of, 84
universal
Σ(Γ)-inverting, 52
T(Γ)-inverting, 53
universal Σ-inverting ring homomorphism, 101
universal R-field
definition of, 105
universal coefficient theorem
for L2 -homology, 77
universal field of fractions
tensor product over orbit category,
34
theorem
Goldie’s, 64
topology
strong, 5
ultraweak, 5
weak, 5
Tor-dimension, 6
torsion
module, 19
125
definition of, 105
of a semifir, 75, 106
recognizing, 107
universal localization
K-theory of, 76
and K-theory, 85
of a commutative ring, 102
unreduced L2 -homology, 24
very trivial representation, 40
virtual Euler characteristic, 28
virtually abelian group, 81
virtually cyclic group, 36
von Neumann algebra, 5
K0 of, 81
K1 of, 84
amenable, 45
center of, 81
finite, 5
of a group, 1
of the group Z, 3
of type In , 81
of virtually abelian group, 81
type of, 81
von Neumann regular ring, 6, 11,
29, 114
and chain conditions, 96
and directed union, 71, 95
and matrix ring, 95
and zerodivisors, 96
center of, 95
definition of, 95
is division closed, 103
is rationally closed, 103
weak dimension, 6
weak isomorphism, 7, 83
weak topology, 5
zero divisor conjecture, 28
zerodivisor, 4, 98
conjecture, 28
in von Neumann regular ring,
96
126
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Lebenslauf:
geboren am
Eltern
26.2.1969
in Uelzen
Gerhard Reich und Dagmar
Reich, geb. Gratzfeld
1975-1979
1979-1988
16. Mai 1988
Grundschule in Uelzen
Herzog-Ernst-Gymnasium
Uelzen
Abitur
Juli 1988 - Oktober 1989
Wehrdienst in Lüneburg
1989 - 1994
Studium
der
Mathematik
mit Nebenfach theoretischer
Physik an der Georg-AugustUniversität Göttingen
Vordiplom
Diplom
Schulausbildung
Wehrdienst
Studium
22. April 1991
27. Oktober 1994
Beschäftigungen
Oktober 1994 - April 1995
ab Mai 1995
133
wissenschaftlicher Mitarbeiter
am SFB 170 (Geometrie und
Analysis), Göttingen
wissenschaftlicher Mitarbeiter
am Mathematischen Institut
der Universität Göttingen